Hi jenndaisy82, welcome to the forum. Given that you're interested in school eventually, the first area that pops into mind is the Nob Hill area. It has everything else you've mentioned and you'd be minutes from UNM campus. There are other schools around town (CNM, etc...) but none are so obviously bi-located with other cultural/social stuff. Only thing is that it's going to carry some of the heftier price/lease/rents in town. They just finished some new lofts right on Central. Walking distance to everything you mentioned and some, but no tellin' what they're going for.
As for the bike ride part, while Albuquerque loves to brag about it's bike friendly status, as a biker I can tell you that for the most part you're aware that you're sharing space on a grid system meant solely for cars. This means often being separated from multi ton SUV's being piloted by people busy texting on their cell phones by a 6" stripe of paint. There ARE some amazing dedicated paths/trails around town, but you'll rarely be able to stay on them and get where you're going.
Search: Nob Hill and see what pops up. If it's a bit pricey, let us know what a comfortable range would be and we can adjust our recommendations accordingly.