Good call, public transport will get you between the main towns, but the real magic, like Song Kol or the passes south of Issyk-Kul, is way off the beaten track, and you simply cannot reach it on a bus. A sturdy 4x4 with proper ground clearance is the way to go, because the asphalt can disappear pretty fast once you leave the highway. Bring your passport, your home driving licence, and an IDP is a sensible backup to carry, even if not everyone asks for it. Check out the listings on this site to browse your options and rent a car in Kyrgyzstan
https://getrentacar.com/en-US/kyrgyzstan .