Installed new starter. Has power. Turn key and it won't start and lose power. If I remove & reinstall battery cable the power is back but it starter still won't work

Battery good, cables good, all connections clean and tight, no corrosion on battery terminals?
Try to determine if click is from solenoid on the starter or starter relay. Turn the headlights on and watch them while someone turns the key to start. If they get dim, probably the starter-have it tested-even new ones are sometimes bad. If lights don't dim, you will need to check the start circuit. Check for power on the small wire to starter solenoid when key is held in start. If no power there, work back towards the ignition switch. Check the neutral safety switch-should have power in and power out when key held in crank. The ignition switch will have a start wire out of the switch going to the neutral safety switch. It should have power with key in start, the only time the start wire will have power.
Post back with what you find-good luck.

What happens when you turn the key to on? Do the warning lights and gauges come on? Turning the key to start, do the lights and gauges stay on? Just nothing at the starter?

Or does power just shut off when you turn the key to start, everything goes out, no warning lights or gauges? Then check cables, battery terminals, grounds for clean tight connections. There is a problem between the battery and the ignition switch.

If the switch still receives and sends power out (to the dash, for one) but start circuit is not working (from battery to ignition switch to clutch switch to starter relay to starter solenoid to starter to ground-and back to battery), then here's where we at: with key off, use a small screwdriver and short the big cable at starter to the starter solenoid small wire connection, or use a small jumper wire to do the same-battery cable at starter to the small wire connection on the starter solenoid. Starter motor should spin and crank engine. Does it? If so, start checking the start wire out of ignition switch for power when key is held in start. If you have a clutch switch, check for power in with key in start. Should have battery voltage to the clutch switch with key held in start. Keep checking the start wire along the start circuit. Not sure how the start relay is wired into the circuit, would have to see a diagram, but if you pull the relay out and use a test light or multi meter, could check for power on the terminals where relay plugs in. Key off, should have one terminal that is hot. With key held in start, at least one other terminal should also be hot. This would be the start wire out of ignition switch sending a signal to the relay.
Good luck, hope you find it. Sorry I don't have a wiring diagram to help you more.

Check to make sure you are getting voltage to the small post on the starter solenoid when the key is turned to the "crank" position. if so, check all of you connections on the battery cables. If your connections are all good, your cables are in good shape, and you still do not get the starter to crank, I would have the starter removed and tested.

You would need to take a meter or test light and check for power after the battery. You could have a bad connection inside one of the cables.
There should be 12 volts at the battery post and at the cable end on the starter. Also be sure and check the ground connections.

Are you sure your battery is good and able to take and keep a full charge? The only thing in the start circuit that could cause a big drain on the battery is a shorted starter. But if you can send power and it cranks okay, I don't think you have a bad starter.

Is the ignition key on when you crank it and have no spark? Does your distributor rotor turn when the engine is cranking?

I would try a new solenoid. If it clicks it should send power on to the starter on the big cable. Make sure all battery and ground connections are clean and tight. If still a problem, buy a voltmeter and learn how to do voltage drop tests. This will help find bad connections.

CHECK BATTERY VOLTAGE SHOULD BE 12.6 VOLTS. IF BATTERY CABLES TIGHT AND BATTERY CABLES AND BATTERY POST CLEAN FREE FROM CORROSION. STARTER COULD BE BAD.TAP STARTER SEE IF IT CRANKS. IF NO REMOVE STARTER HAVE IT BENCH TEST. TURN OFF RADIO BEFORE DISCONNECTING NEGATIVE BATTERY CABLES TO KEEP FROM LOSING RADIO CODE.

TEST POWER LEADS TO STARTER FOR POWER
AT STARTER (BATTERY POSITIVE TO STARTER BATTERY NEGATIVE TO ENGINE BLOCK)
THEN TEST FOR SIGNAL POWER TO SELENOIDE ON STARTER
(KEY TURNED TO START POWER SHOULD GO TO SELENOIDE)
IF NO POWER TO SIGNAL WIRE CHECK FUSES AND RELAYS
ON STARTER CIRCUIT
IF THIS CHECKS OUT MAY BE AUTO TRANSMISSION NUTRAL PARK SWITCH
TRY TO START IN NUTRAL OR WHILE SHIFTING TRUE GEARS SLOWLY WITH SHIFTER
IF STANDARED CHECK NUTRAL SAFETY SWITCH

CHECK THE INSTRUMENT CLUSTER FOR POWER TO GAUGES TO SEE IF THEY WORK WHILE IGNITION IS IN THE ON POSITION.IF NO CHECK THE 40 AMP IGNITION FUSE.IF FUSE GOOD IGNITION SWITCH IS BAD OR HAS A LOOSE OR BROKEN WIRE.IF FUSE GOOD ALL GAUGES IS WORKING.I BE LOOKING AT STARTER AND STARTER SOLENOID.REMOVE THEM HAVE AUTO ZONE TEST STARTER.OR YOU CAN GET A REMOTE STARTER SWITCH AND SET OF BATTERY CABLES.HOOK BATTERY CABLES TO A BATTERY.WITH STARTER OUT OF CAR TAKE BATTERY CABLES HOOK NEGATIVE BATTERY TO STARTER BODY FOR GOOD GROUND.TAKE POSITIVE BATTERY CABLE HOOK TO STARTER POSITIVE TERMINAL WHERE POSITIVE BATTERY CABLE GOES.THEN HOOK THE REMOTE SWITCH RED WIRE TO STARTER POSITIVE TERMINAL AND THE OTHER REMOTE SWITCH WIRE TO THE S OR SWITCH TERMINAL ON STARTER SOLENOID.PRESS THE REMOTE SWITCH IS STARTER RUNS GOOD WITH NO FUNNY NOISES ITS GOOD IF STARTER DONT RUN THE STARTER SOLENOID OR STARTER IS BAD. THE PROBLEMS IN YOUR CAR COULD BE A BAD KEY TRY ANOTHER KEY YOUR CAR HAS THE THEFT DETERRENT MODULE SYSTEM BEST TO HAVE CAR SCAN TO CHECK THE THEFT SECURITY MODULE AND THE TRANSMISSION NEUTRAL SAFETY SWITCH IF ITS BAD STARTER WONT WORK.ALSO CHECK THE THEFT DETERRENT MODULE RELAY. BEFORE REMOVING BATTERY CABLES TURN OFF RADIO SO YOU WONT LOSE YOUR RADIO PLAYING CODE.

Hello.
When you installed the new starter, you may have forgotten to tighten the positive and negative cables going to the starter giving it no power what so ever.
A second solution could be when your removed the old battery, you popped a few fuses that your were unaware of. Try taking a look at the fuse box/panel and see if you can find any popped fuses. A bad fuse can result in no power to certain parts of the car.
One more solution can be that the positive ground that goes to the battery, trace it down at the other end, not the battery end, but the end that should run into a box and have a bolt, make sure that bolt holding the opposite end of the battery is tight, a bad connection can do that also. Do the same for the negative if all else fails.
Hope this helps!

check the ground to the engine, make sure the terminals are clean and tight on the starter and on the battery and have the battery tested.
24 months ago for a rebuilt starter is about the lifetime you could expect.
Did u change the battery cos' it was getting harder to turn the engine?
If the starter gets lazy (often a loss of windings due to varnish degradation) and hence a drop in power when activated. This wears batteries out and eventually the starter will fail. A lazy starter has that labored wuh wuh wuh sound instead of a faster regular sound
If you odnt find bad cables replace the starter.