Tips for the Dungeon:
1 - Sticky bombs. You'll want to get rid of as many of those spikes as possible. Especially in Master mode, you'll come across times where your only option is to dodge or take cover if your health gets low.
2 - A hook as well as some sort of flying. This ties in with #1. It will help you get to safety.
3 - Platforms and Dangersense potions. There is now a new danger in the Dungeon. Breakable floors which often times lead down to a drop filled with a pit of spikes. Dangersense potion will alert you to the blocks, highlighted in pink. Break them then build a bridge across using platforms.
4 - Be aware of your surroundings. Try not to get confined into a small space. Those flying skulls can hit you with a no action debuff which can lead to your demise and its tough to dodge them when in a small area.
5 - Banners. Sometimes patience is the best bet in certain areas. Keep killing the Angry Bones until you get a Banner to help you. Then place the Banner and collect some more to place in new areas when you run into trouble.
6 - I prefer potions of Restoration over Healing Potions. You get 10 less health but the reduced time inbetween potion uses helps and makes up for it.
As far as Armor goes beef up as best you can and if that isn't enough throw in some Ironskin/Endurance potions. For weapons most types work fine except for prehardmode summoner (unless you play classic). Be sure to include something with good knockback as crowd control is essential. Don't pay attention to the tooltip as some weapons with low knockback per the tooltip work very well. Have at least an extra weapon that stops enemies from getting closer to you.
Remember that the Mages projectile can be stopped by attacking it. Even using a pickaxe with timing you can nullify the bolts they send at you.