Central Tucson is actually pretty convenient for some of the good rides in town, namely Gates Pass, plus the Shootout leaves from the University area nearby. If you’re looking for cultural offerings, I’d look for a place near the university / 4th ave area or south of downtown in the barrio viejo or armory park areas. Really anywhere in a few mile radius of the university will be fine. Note that these aren’t generally luxurious areas, but are authentically Tucson. If you’re there purely for cycling, like more suburban and new-build accommodations, and aren’t too interested in being close to bars/restaurants/etc you may prefer to stay in/near Starr Pass west of town or in the foothills of the Catalinas.