Starter

Make sure battery has a full charge and battery connections all good. There are two voltage circuits at the starter motor, one comes from the battery, hot all the time. The other circuit goes hot when starter relay energized. You can use a test light to check for voltage. The starter motor usually grounds through the engine block. Any testing at starter motor, make sure tranny is in park or neutral and parking brake is set. Note the operational info. Your security system could be effecting your starter motor?

It is important to note any little issues also, such as if you hear a click from the starter relay or solenoid when you attempt to start the car. Info such as if your starter worked perfectly or had been dragging recently is useful in diagnosis. Auto Trans or Manual? Did you have your old starter tested before buying a new one at the parts store and if so, what was the verdict? Also, try another key to the ignition that you know has worked before, chips can be damaged. I say this due to the "chip" in the key and a possible malfunction in the cars registry. Additionally, try starting the car in Neutral, just to test the Neutral safety switch. Back to the starter situation. If it's not the key and your battery is fully charged, you'll need to do basic troubleshooting to find out where the voltage stops. ABSOLUTELY make sure you have a fully charged battery, then verify the connections are clean and that voltage is passing through them by using a test light at the battery cable connection, and, at the starter connection. Also make sure the ground connection is clean and tight and verify that you are getting a ground to the engine by using your test light. If you have a good 12+ Volts coming to the starter, the solenoid would be your next test. You can bypass the ignition system here and apply 12V to the solenoid to see if everything here is working properly (as long as you know where to connect the 12V properly). If it is getting 12V to the system and the starter works when you apply voltage to the 12V wire going to the solenoid, then you'll need to back track by testing the wiring dealing with the entire starting system. I know it's a lot, but, you're dealing with the Ignition system and the starting system, Good luck:)