1996 Chevy Cavalier starter Question

Bluelightnin6 is right.
Nothing to do with starter, one without knowing cars much can tell, unless it is a design for both starter and alternator (generator).
If the charging circuitry is in good shape, it will be the alternator, w/o current generated from the alternator, your car draws current from the battery until it goes flat.
My damn GM cavalier got me replaced alternator twice within 6 months because the damn Delphi supplied crap alternators. To buy one is relatively cheap (not really cheap) but to install w/o tools, you can release the tension arm that let the belt runs thru. My damn cavalier gave me a lot of problems and costed me much good money after this bad stuff, I just return home with it on a 2nd coolant temperature sensor replaced, it is one of the culprits that caused NO START immediately after use.

All those other replies to your question sound like good advice to me. Sounds to me like you need a new alternator, unless you got a faulty new battery. That is unlikely that you would get a faulty new battery, but it did actually happen to me once. My advice to you is to go down to your nearest battery shop and get them to test your battery and your alternator to see which one is giving you the problem. They will usually do both of those tests for free unlike most garages or car shops. That should narrow down the problem then you can fix it yourself from there. If it is the alternator and I'm thinking that is what the problem is, then you just need to go get an alternator and replace the old one that can be used for a doorstop. Its very easy to replace it if you do not know how. You just unplug the electrical wires from the old one then loosen the tension on the belt that drives the alternator then simply unbolt the old one and bolt in the new one. Then just retension the belt the same tension that it was at before you took out the old alternator. Also a good rule of thumb is to tension it so that you can push your thumb in on the belt about a half inch or about 1.25 centimetres.