Car wont stat...battery voltage good and also good connections...will start after a chage or jump start but will not crank back up when turned off...starter solenoid clicks

Does the car push-start (or actually the engine crank turning)? If yes then it has to be a battery or starter solenoid/starter motor problem. In some cases it can be the power cable to the starter motor. Check the terminal connections to be clean, good and solid. First of all though check the voltage drop at your battery when the key turns to start. If the voltage drops under 10V then your battery is weak and you should try either jump starting or put another known good battery and test.

You have to make sure battery has a full charge and that means testing. Your saying the starter motor won't even crank the engine? I'd have to check starter wiring circuits for voltage drop?Also, make sure battery connections are good. Engine cranking, your systems get battery voltage. Engine running, the vehicle system get generator voltage which can be several volts higher than battery voltage.
Use a digital multimeter across battery posts, everything off, what is the reading, the generic spec for a full charge is around 12.6 volts. Leave the meter leads hooked up, have helper turn key to crank, what is the reading, how far does it drop? The generic spec says it shouldn't drop below 9.6 volts, that spec may be subjective but, close.

Please provide with more detail. Does it make a click sound when turning the ignition? Does it crank?
Have you tried to jump the battery with jumping cable or tried it on a know good batteries?
- If the car does make a cranking noised then the starter is good but weak battery.
- If there is no noise or clicking, then dead battery.
- If you can hear the clicking when turning the key to start, then check for battery GROUND.
- If battery were tested and it read 12.5 - 14.0 DCV. Then your starter are bad.

There also other thing to check like security alarm system with ignition switch kill... But most common problems with fail to start are mention above.

make sure is ither neutral N or P. it looks like you have en electrical problem check ing. switch check for a blown fuse.

The starter is bad if you can't jump it
from the battery

Take it out & test it

If you just hear a click, try this:
This procedure usually works:
·Check the battery cable connections:
Make sure the positive cable, makes a good connection at the battery and at the starter. Make sure the negative cable makes a good ground connection at the battery and (preferably) at the engine.

·Check the battery the simple way, like this:
Turn on the headlights, then try to start the car.If the headlights do not dim or go out, then the battery is likely ok.

·Check the starter and solenoid:
If the wiring looks ok, then look at the starter solenoid for a good connection, and proper voltage at the starter(10 volts minimum while cranking).
If the starter spins and no crank, the problem is most likely the Solenoid's bendix gear is stuck.
Have someone hit (not too hard) the starter while you try to start the car. This usually works by dislodging a stuck bendix gear.
·Make a simple test of the alternator:
If you can, somehow get the engine running, measure the voltage at the battery. It should be at least 13.6 volts to properly charge the battery.

If the car doesn't start with a jump you might have a bad alternator. The alternator supplies voltage to the car once the engine is running. If the battery is dead, and the alternator isn't putting out sufficient voltage, the engine cannot run. Then again sometimes a car with an extremely dead or internally shorted battery WON'T start with a jump. Replace the battery and try again.

will it start with a jump? check battery for full charge, maybe alternator is not charging, check battery connections all the way down to starter, check selonoid by cross jumping terminals to engage starter.

Could be the cables or the starter. Remove the starter and have it tested.

is the battery fully charged? will it jump start/ if the battery is good and it wont jump start. It sounds like a bad starter, sometimes on a GM auto you can tap the starter housing with a hammer and get it to start if this works you will need to get a new starter. Note tap the starter dont beat on it.