Car won't start

I have a 1998 Saturn SL2 series car. Last month I replaced the battery and last week when I went out to start it all it would do is give off a series of rapid fire clicks. The radio and dash lights all worked but the engine would not turn over. The next day I went out to try to start it again and it was totally dead-no lights, no radio, no dash lights. This happened right after we had our first cold night. For about a week or two I'd been able to get it started but it did give off 2-3 of those rapid fire clicks when I turned the key. I am unsure if the battery was not good to start with and the cold weather stressed it or if I'm having problems with the ignition switch, or the starter or something else. I would appreciate it if you could point me in the right direction. Thanks.