I have a Half-Elf Level 5 Wizard character, who is quite the pyromaniac. His school is that of Illusion. I'd like to find a way to make him fire-resistant, in such a way that it will be persistent. Being a caster, armor's not really the best option, and I'd prefer to avoid multiclassing if possible. What options do I have?
Spell wise, at 5th level you can get absorb elements. It doesn't give persistent fire resistance, but it can get you fire resistance when you need it using a reaction.
Fire shield can also get you resistance, though again, it's not permanent. On the up side, it will not use your concentration.
The only way to get it persistant with your character within your parameters is getting a magic item. You mention not wanting armor, but there are other items that would do the trick. A frost brand sword, helm of brilliance, mask of the dragon queen, piwafwi of fire resistance, ring of fire elemental command, ring of resistance, staff of fire or a shard of the ise rune.
Hope that helps.