1996 Nissan Sentra Won't Start

1996 Nissan Sentra GXE 100,000 miles. For about four years now my sentra has had the habit of not starting. At first it would happen ever 8 or 9 months but as of a month ago it has been happening a few times a week. I have always thought it was the battery but recently I bought a new battery so now I know it isn't the battery. When I put the key in the ignition the car just clicks. It doesn't try and rev up or anything. I get nothing. Sometimes if I just let it sit and go back and try it, it just starts up. But just tonight I had to leave the darn thing at work because it would not start. The "check engine" light came on a week ago when it wouldn't start. The radio, lights, and heat work just fine.

I also notice about 3-4 months ago the car alarm I have on my car start going mad. By that I mean, the car would set the alarm on it's own without me pressing anything. Sometimes the car would disarm itself without any command from me and then it would later rearm itself. I've had the car alarm for about 10 years so I know it in and out; there is no timer mechanism I'm missing. Also, the thingy that tells you how many miles are on it stopped working over a year ago. It moves occassionally but isn't consistent. I'm a chic and I know how to change my tires, air filter and stuff, but I don't know anything about cars other than small maintanence. Does anyone have any ideas what's going on? I had the car looked at 2 months ago and was told I needed a tune-up. I paid for that and the darn thing still doesn't start. Is it the starter, the altanator, what? Thanks in advance.