troubleshooting steps
how soon after getting the new battery did the problems start? if you still have the old one, swap it back and see what happens. when you hook it up with jumper cables, does it always start? i mean 100% of the time. if so, then it's most likely not your starter or even getting jumped wouldnt help and it also eliminates the starter solenoid, fuel pump, clutch lockout switch, and ignition switch.. do your lights die when you try starting it? or dim? if they do, then its the battery. you couldve gotten a defective one. try hooking up a pair of jumper cables to your car, but dont connect them to another car. just put your side on and see what happens. if that works, your battery cable connectors are worn out and not making good contact. very cheap. happened to a friend of mine. does your battery light ever come on while driving? if you have any doubt about your alternator, take it to autozone, advanced, o'reilly's and they should be able to test it for free. theres nothing else i can think off right now where the car wouldnt even turn over, but starts just fine when jumped.
Moses