1999 Chevy Suburban Starter and Ignition

I have a 99 suburban. The starter went bad, just clicked when turned key. I replaced the starter with a new one. When I hooked up the ground wire on the battery, the starter started turning the engine over. The key was not in the ignition. I took the starter back and had it tested because, I thought it must have a faulty solenoid, but it tested ok. I put the starter back on and hooked up the ground wire to the battery. Again the starter immediately started turning over the engine. I then turned the key on and hooked up the battery to see if it woudl start the truck and it turned the engine over, tried to start but would not. Almost like it was not getting enough fuel. I have looked for loose wires etc and checked fuses. Could the problem be the ignition?