Car won't start

I have a 2002 Saturn SL2 and I live in PA. The last time it got warm out, I had trouble that the car wouldn't start. At that time (about 1 month ago) I replaced the battery (I'm not sure that was the cause but a 3 year old battery is usually suspect).

Now the car won't start again. When I turn on the key, I get no engine noise. I occasionally get some lights on the dash & when the key is in acc, I can't wind up the electric windows. The results for turning the key are very sporadic. Sometimes I get a security light and check engine light & that's all, sometimes I get full light panel and sometimes I get nothing. Anyway, I get no engine noise.

A little more background:
When we changed the battery, there was alot of corrosion around the terminal (a side mount that you screw into). I got the corrosion goop that they sell with a new battery & the guy said to dump it in the hole. I also bought a new screw-in cable connector for the positive post. the guy who put the battery in did not clean the connector from the car of corrosion. Could corrosion do all of this problem? Unfortunately, it always happens away from home where we can't address it in our own driveway.