1996 Blazer No Start

Hi Everyone
My parents have a 96 Blazer with around 80k miles. The thing was having some starting issues which at first seemed like a battery problem. I went and had the battery tested and was told that it needed replacing. So I bought the new battery put it in and it didn't start on the first try, just a click from the starter. It started on the second time. It was intermittently doing this for about 4 days or so and now it won't start. Its sitting in a parking lot until I figure out what to do. It appears that the starter is the problem but I'm not sure. First where is the starter solenoid? I hit the starter while my dad had the key in the run position. How would I go about testing the solenoid? It would be nice to at least figure out the problem even if I have to have it towed. The starter only makes one click but nothing else. Battery is new so I'm figuring that is not my problem. Any advice would be appreciated.
Thanks-
Mike

The solenoid is on top of the starter. It would be best to take if off the vehicle and take and have it tested.

Ok here is an update-
Had it towed home and I Took off the starter and alternator and had them tested at two places and both times they passed. I am at a total loss with this thing. The new battery was run down which made me think it was the alternator. Is it possible that the new battery could be a bad battery? It has to be possible right? Does anyone have any suggestions. I really don't want to take it somewhere only to find out it was something stupid.
Thanks
Mike

A bad starter could draw some current and run down a battery overnight, but it may be the battery is running down from the attempts to start it. First If I understand the problem there is a click and the motor will not "turnover" or spin to try and start ? If thats the case then it could be the Starter, Battery, or connections to them. Since the starter tested good I would get the Battery fully charged and then clean all connections and grounds on the cables for the battery and starter and see if that helps. One thing to watch out for is if it does start cranking but will not fire it could be the battery voltage is low from the attempts to start it. If the Battery voltage is low the electric fuel pump will not build enough fuel pressure to start it. First you need to get it to crank properly and you should be all set. I forget how much room is near the starter but you can directly apply 12 volts from the battery to the starter and see if it turns the motor, that would help tell you if its a connection problem or if smething strange is going on with the mounting of the starter. One more quick check is to try and turn the crankshaft with a socket or some times you can turn it by grabbing and trying to spin the serptine belt to make sure the motro can spin ok and is not locked up.

Another Update-
Took a gamble and replaced the starter and that fixed my problem.
THanks for the help
Mike