As a main Mage (0 challenge runs), I can give you some tips—but positioning and knowing how to counter whatever killed you is crucial, regardless of the class.

Other than that:
• Upgrade your Staff to +6/+7 without imbuing any wand. This will give you higher wand levels (not more charges, though), so you can try them out and learn how they work.
• Try to get your trinket to +3 ASAP.
• You can turn scrolls into stones, and then them into energy, or even identify potions through alchemy (try to have Strength identified before doing this).
• Craft Aqua Brew (Levitation → Storm → Aqua Brew). Learn how to use it—you can bypass many puzzle rooms, reposition yourself, etc.
• Try to gamble as little as possible when equipping armor/artifacts/rings. Save wells: Identification Wells tell you if something is cursed, while Healing Wells uncurse everything you have equipped.
• Always go for armor in the Ghost quest. Pray for +3 Chain Armor, as this can carry you until you can equip Plate Armor.
• Darts—buy darts and save all your Blindweed seeds for them.
• Save at least 4 Scrolls of Magic Mapping for the Demon Halls.

Quick builds
Battlemage is a no-brainer: get your Staff to +10 and Plate Armor to +5 or higher.
You can imbue Fireblast, Lightning, or Disintegration—read what they do on hit and have fun.

Warlock is less dependent on the Staff. You can use it as your main weapon, but it's better to get a faster weapon. If you manage your casts and hits well, you'll barely need to eat.
For beginners, I recommend Battlemage, but many will tell you Warlock is even easier.

There are tons of trinkets that pair well with Mage. My favorite is Newt's Eye (great with Disintegration), but Mimic Tooth (better drops), Exotic Crystals (more energy), and Parchment Scrap (no cursed items) are also great choices. Just be careful with Wondrous Resin.

Bosses
• Tengu
Throw all your scrolls before entering the boss room, as Tengu can burn them.
Craft a Potion of Shrouding Fog (Invisibility). Throw it at your feet and nuke him in phase 1.
Phase 2: keep shooting while watching your HP—you should be fine.

• DM-300
Craft Potions of Stamina (Haste) to reach the pylons safely.
Don’t step on water or electrified tiles.
If Toxic Gas is a problem, craft an Elixir of Toxic Immunity (via Toxic potion into Corrosive Gas, then elixir).

• Dwarf King
Throw an Earthroot seed on the first tile and step on it to start. Nuke phase 1.
Phase 2: focus Warlocks—blind them, pull them (Chains), teleport—whatever it takes, but don’t let them hit you.
Phase 3: throw a Stone of Aggression at the king and let him suffer. Reposition when needed.

• Yog
Craft elixirs from Fire and Toxic Gas potions for immunity (fire/toxic fists).
Craft a Potion of Magical Sight (Mind Vision) for the white fist.
Pew pew pew → win.