Intro/Mines

Pick up the Halo, then head straight to the Mines. It doesn't matter how fast you do this part, since you're bound by an elevator timer at the start of the Mines. Just make sure to survive the boulders.
Upon entering the Mines, drop down one screen on the left side, kill the goblin, then kill the snake on the right side with the bow. Wait for the elevator and ride it up, and take the downtime to arrange your hotbar and screw around a bit. Pick up the Fire magic quest, then head back to the elevator shaft, and drop down two screens to the right. Kill the blacksmith at the top, pick up his bellows, the anvil, and the green weapon from the chest. Head down one screen, clear out all the goblins, pick up the blacksmithing hammer. Then b-line to the boss.
On Dynidon, switch to the blacksmithing hammer. Just after he starts his attack, jump and swing, then swing again when you land; with perfect timing, you can hit him twice per attack. Either way, he should die in one attack cycle.
You will need as much good drops from the mandatory kills in this area as possible. You might consider killing excess goblins for more drops, but it may not be worth the risk.

Shopping/Sewers

Head to Zoran. Buy 1 Heat scroll, 5 Zoran teleport scrolls, 4 Return teleport scrolls, and as many Phase Door scrolls as you can "afford" (between 1 and 3, depending on the sell value of drops you've gotten). Destroy all the barrels in this screen and loot both chests, then head to the Cook. There, buy 50 blue essences and one Cook teleport scroll.
Head towards the Sewers. While fighting Momoku, you can hit 4 arrows per phase, then use the downtime to arrange your scrolls onto your hotbar without losing time. Pray to RNGsus that a thief doesn't steal your 40 coins. In the next room, gather the final pieces of wood and, if necessary, enough experience for level 2. Also pick up all Wands of Frostbolt on your way. Proceed to the west entrance of the sewers; use phase doors to save time here. Pick up the Ice magic quest and the ferns quest, then go into the store and buy a Fireball tome (if you can't afford it, an alternative is to buy a teleport to sewers scroll and pick it up later).
Cast Return to the ferns quest, and head for the eastern entrance (again picking up all wands on route). After the first room of goblins, take a detour left into the room with crows to pick up a unique sword (hidden in the walls on the left), then head straight to the east and pick up the Trident. Proceed to the sewers, pick up and complete the ice elemental quest using one wand of frostbolts. Summon your new pet, then head to the boss area without falling down anywhere.
In the volcano area, use a new wand of frostbolts to clear any enemies that get in your way. Don't bother breaking them for loot. When you arrive at Drakonis, draw your third wand, position your pet properly, and demolish the guy.

Gardens

This is as good a time as any to hand in the Fire quest (with a Return scroll, ofcourse). Afterward, head to Zoran and stock up on 6 Heat scrolls and 7 Phase Door scrolls, then head to the Gardens. On route to the Gardens, loot another chest with a green weapon, and use 4 of your phase doors to skip the shiny corridor. Head to the gate, making a small detour to pick up three speed potions.
To battle Arborus, pop two of your Speed potions and stab him with your trident until he dies. If he casts his nose beam attack, hide behind your pet. If he casts his insects early in the fight, you might want to use a Heat scroll for safety. The meteor shower is both completely random and totally unavoidable.

Library

Teleport to Zoran and pick up another 10-11 phase doors before you head into the Library, killing every monster conveniently in your way while doing so; you need both the money and the experience at this stage. Be careful never to aggro more than one ogre at a time, they hurt a lot. While here, do the quest for the Slicering halberd using a phase door and a return scroll to bypass the fire. To easily beat Leviathos, summon your pet in the room before him, run to the edge of the platform, then take a few steps back while your pet casts a frostbolt and turn around again. This will park your pet in front of you and make him tank every shot as well as the pulling spell, allowing you to easily pick Leviathos off with firebolt spam.
In the next room, you can easily take care of the skeletons by jumping on the ledge to the left and slowly picking them off with the trident. Skip past the room with the scorpions, beetles and orcs, as it's too risky to do any farming here. After crossing the next room, cast a Heat scroll to get rid of the pesky bats and make going up easier. Finally, open the gate at the top, then head into the shop to buy the Windbreaker bow, a red candle, and a fire robe. Proceed up into the boss room.
Say a prayer to RNGsus, as this boss has a tendency to troll awfully. Before summoning him, though, throw all your potions and teleportation items on the floor. Your trident should make decently fast work of him. Once he's dead, return to Zoran and stock up on all the phase doors you can afford.

Catacombs

Move into the Catacombs, take the lift up to unlock the gate, then drop down and head into the room east of the shop. Phase door past the mobs and head in the door to pick up the Light magic.
Reset the game, then head back to that same room. Stand by the first ladder, cast a Heat scroll, then go down and run around a bit to spawn all the undead. Hang off the ladder out of reach of their attacks, and do some inventory management while you wait for everything to die. You can wait for Thief spawns here if you still need a strange scroll. Repeat the process 2 more times. You should end up at level 6 and with around 20-25 casts of your Light spell.
Once you're done farming Light, drop down to the bottom of the elevator shaft and head right. Phase door across the next room, climb the ladder, get rid of the skeleton wizard, and slip the switch. Then activate the cutscene trigger from the other side, and quickly jump up to unlock the door. Proceed right two more screens, pass through the door, then use phase scrolls to travel through the blood river towards the boss.
Sux Mortis is very straight-forward. Just dodge all his attacks while casting Light at it at every opportunity. Be aware that after his death animation, his scythes can still hurt you.

Lab/Prison

Teleport to the Cook, and spend all your money on red essences (around 700 of them). Any remaining money can be spent on phase doors. Teleport to the Gardens and make your way to the Laboratory. The 'hanging gardens' room can be easily cleared with use of a heat scroll, and doing so allows a safer (and potentially faster) entry into the Lab: hang off the ladder, wait for the scientist to hit your heat aura and turn around, then phase door past him. At best, you'll catch the elevator cycle; at worst, you can position your pet to take his aggro when you do get on the elevator, making this room much safer.
B-line straight to the prison, and straight to the boss room. Stand on the right edge of the platform and barrage Gauntlik with fireballs, then get in position for whichever attack he's chosen: for rock, stay at the edge and move as little as possible while avoiding his attack; for paper, stand in the middle and run along the center lasers; for scissors, stand in the center, then as soon as the first slice sounds, phase door behind him and start attacking him. Avoid using fireballs while he's still close, as you can easily kill yourself, and any damage you take will stop the next fireball cast anyway. If you're lucky, you can kill him in one scissors phase; at worst luck, three phases should do it.

Halls

Stock up on phase doors so you have at least 25, and make sure you have one heat scroll left. Then head into the Halls. Almost all screens in the Halls can be crossed using phase doors. The exceptions are the first two rooms; the room with the gate loses you time, while the room after that has a spellcaster that interrupts you. If you're unlucky you can get hit by a slime while going down a ladder; if you do, unequip your weapon quickly, then just continue. Head into Medeox's lair.
After summoning Medeox, climb up to the top-right platform. Set your aim to the fifth snake from the left, then spam fireballs until all his snakes are dead (reaim when necessary). Then aim at Medeox himself, and barrage him with fireballs. Always be mindful that he might rise up to your platform, allowing his eyebeam to instakill you. Turn to the left when he does that attack, and if he's near half hp, take the time to use a heat scroll; it'll take care of the ghosts he summons in his second phase.

Tower

After Medeox dies, head up to the Library and pass on into the Tower. To make matters easier on yourself, use a heat scroll to clear the room to the Ring of the Skies, that allows you to jump further; it eliminates all the pixel-perfect jumps from here onward. Proceed up the tower, avoiding contact with any ghosts; if they steal your trident, you'll be sad, and if they steal your Slicering, you're gonna have to reset. Very carefully head up to the Tower shop; you want to kill all (or most of) the spiders in these three rooms, but they're very likely to screw you over as they fall onto you from the darkness. The best way to handle them is to stand on the edge of a platform, so they won't fall onto you, but down past you.
Once you make it to the shop, buy an Invoke Fire Spirit tome, and stock up on arrows if necessary. Proceed to the right to clear out more spiders, then down to unlock the gate. Reset the game, and continue farming spiders until you're level 10. Once you are, head left and down so you can avoid having to pass the ghosts again, and enter the Wizards' Tower.

Wizards Tower

The wizard tower can best be summarized in terms of strategies when dealing with the monsters:

• Blue Wizard + Eye: Wait until the wizard faces away from you, then phase door past them. You can attempt to shoot the eye before it Slows you, but success is inconsistent.
• Blue Wizard: Run past them, they're not very dangerous.
• Red Wizard: Kill him, let your pet kill his fire elementals.
• Gelitanious Cube: Phase door past it. Resummon your pet before it dies to the DoT.
• Purple Wizard: Stand underneath its platform, clear any skeletons that are in the way, cast fire spirits, wait patiently.

For the room with two gelitanious cubes, you'll want to pull them back a small distance before using the phase door, so that you can use the ladder to get past them.

Elven Lands

As soon as you get control of the satyr, climb up the bottom ladder and hang on it in such a way that your head is hidden behind the platform; enemies won't be able to see you, so you don't have to worry about this area yet.
Learn which mobs can be skipped and which are best picked off before they become a threat. On Easy, many elves won't aggro you immediately, allowing you to easily pick off problematic ones. Rely mostly on the Slicering halberd and fireballs. Without the Ring of the Skies, almost every jump in this area is pixel-perfect, good luck if you insist not to use it.
After destroying the crystals, return to the satyr, jump off the ladder, then use the deathmyst spell to clear out the horde of monsters that's gathered up.

Sewers

Before picking up the ferns quest, head on left to pick up the Ice magic quest. Then, before doing the ice elemental quest, proceed right to loot the trident. This trident is mostly identical to the weapon bought in the Catacombs, making the sequence break unnecessary.

Gardens

To kill Arborus, use two of the Speed potions you've picked up before in order to make your trident attack speed proper. It will make you attack just as fast as the infinite attacking glitch would have.

Library

Rather than a bow, use the Firebolt spell to defeat Leviathos, using the same technique to do so. Use the money you have banked here to buy a robe and candle; fireball will be your main source of damage against bosses. Proceed to the boss room. Drop all your teleportation items and potions on the floor, specifically putting healing/regeneration potions in the center of the room, so you can pick them up after landing a hit and quickly drink them when needed. The trident should be strong enough to make (relatively) short work of him.

Catacombs

On route to the boss room, on the first screen to the right of the elevator, phase door to the ladder and kill the one skeleton wizard, activate the switch to spawn the platform, then trigger the cutscene and quickly jump onto the platform afterward, open the lock during damage invulnerability and run out again. Then proceed as usual.

Tower

Make a small detour to pick up the jumping distance ring, using a heat scroll to make the room easy.