How can I tell what version of Linux I'm using?

Often times I will ssh into a new client's box to make changes to their website configuration without knowing much about the server configuration. I have seen a few ways to get information about the system you're using, but are there some standard commands to tell me what version of Unix/Linux I'm on and basic system information (like if it is a 64-bit system or not), and that sort of thing? Basically, if you just logged into a box and didn't know anything about it, what things would you check out and what commands would you use to do it?