If you happen to be in town during Naadam the Mongolian summer festival, usually in early July there's a nice little celebration outside of town up on the hill, with horse racing, drinking, and lots of people picnicking and shopping at tents set up there.
Most people come to see the surrounding areas and don't spend a lot of time in Ãlgii itself. It really helps to know someone local who can introduce you to other people! Learning at least a few words of Kazakh will also be helpful and people will really appreciate it.