What's cheap?
Cocktails, go to Hummingbird on Bath Street. Good cocktails, nice bar (though can be busy at weekends) and not a bad price. I'm pretty sure it also does food upstairs, though to me it is a bar and club rather than a restaurant. If you want the best cocktails in the city, go to The Blue Dog. However, they are around £8 a drink. Not too bad, but it's expensive for Glasgow.
American type cuisine. Got me a bit stumped there. You after burgers/ribs that sort of thing? Or steak? It's a bit hard giving you suggestions without knowing your definition of cheap and what you're after really.
Lucky 7 Canteen may be worth a look, and then you're on the best bar street in Glasgow for drinks afterwards. Food is dirt cheap, drinks are reasonable and it's good fun. Restaurant one side, and then a busy but cool bar the other.
I've heard good things about Ad Lib, though haven't eaten there myself.
Ketchup on Ashton Lane is pretty good, but I think it's more of a lunch place than somewhere for dinner but I prefer nice sit down restaurants. After that you'll be on Ashton Lane which is really cool and full of bars with quite a good buzz at night.
If you're wanting steak, and willing to up the budget then The Butchershop Bar & Grill is the best in Glasgow. The meat is amazing, as are the cocktails. However, me and my mother spent around £150 last time we were there but we both like a drink.