To meet people and see traditional nomadic culture in a more 'authentic' setting, you should go to small villages, bazars, animal markets, or stay with families in small guesthouses. Don't expect to see much authentic local life in the more touristy areas or on the more popular Kyrgyz treks. There are good day hikes and local interactions around Temir-Kanat and Chon Kemin in the Lake Issyk-Kul area, also around Lake Son-Kul. There are easy, short (around 3 day) treks in that area too.
In terms of timings, the herdsmen are in the mountains between the end of May and the beginning of September. It is hard to say where exactly, as they move around. And in August there are festivals almost every weekend that can be included.
And if you're interested in nomadic culture, I'd highly recommend a visit to the Nomad museum in Sary Oi. Also visiting one of the festivals held during the summer give a good impression of the nomadic lifestyle.
