Best time to travel is June-September because the weather allows to visit high mountainous areas and experience nomadic lifestyle. The passes are open at that period of time and you can stay in yurts like nomads do.
If you are coming to Kyrgyzstan in October-November, March-May, you have a chance to have a trip around Issyk Kul lake, visit Chon Kemin, Grigorievskoe, Jeti Oguz, Barskoon, Skazka gorges, Tash Rabat caravan sarai, Kochkor village for the felt workshop and sometimes, if the weather is nice, have a day trip to Son Kul lake. You won't see any yurts or nomads but you have a possibility to take a look and cool photos!
It is also possible to visit Bishkek and Osh cities even during winter time as long as you don't mind cold weather.