If you're getting a click when you turn the key to the start position that means the starter solenoid is engaging, which doesn't take much power (a few amps), but the starter isn't turning, which takes a lot of power (100-150amps).
This can be from one, or more, of 3 things;
a weak battery
a dying/dead starter
bad connections.
You've said the battery and starter are good.
You should check your battery ground cable, both ends, not just the battery end, and make sure they both are clean and tight. By both ends I mean also check where it connects to the chassis, not where the cable splits off and goes to the body.
Next check the ground from the starter to the chassis, both ends (when I pus a new starter in mine I got rid of the factory ground and ran the ground to the same point on the chassis that the battery connects to).
Next check the battery positive cable, both ends. Where it connects to the battery, and where it connects to the starter. Make sure both are clean and tight.
Examine all cable lugs to make sure there isn't corrosion between the cable and lug.