02' Cavalier, stutters when idling

I recently bought an 02' Chevy Cavalier, automatic, with 70k miles on it. The car idles at about 6 grand. When I'm stopped the car sometimes stutters and the tachometer either jumps up to 7 grand or down to 5-ish grand briefly, and then goes back to 6. It happens at irregular intervals and I am wondering if this is normal or if there is a potential problem that I should be addressing. It seems to only do it when the a/c is on, but I haven't driven it much without the a/c because I live in Vegas and it's been super hot. The car also sometimes revs (when I give it gas) without engaging and then goes into gear when I'm making left hand turns. It's weird. Thanks for any and all info/help.