It sounds like a connection issue. You probably have a bad connection on your battery ground cable or the positive to the starter terminal. It is a good enough connection to power the lights, but under a heavy load like the starter solenoid, you loos the connection. It could be corrosion, a bad cable, or a loose terminal. Did you loosen the ground cable on the frame to tie in the lights, or is it directly to the battery post? I would start with the ground screw to the frame, then the starter terminal to the battery. What you describe is typical of a bad connection.
I assume the battery connections are good since you put in a new battery, but could be checked as well. Did you clean the battery posts/cables with a wire brush battery cleaner before installing? Probably need to do on a tractor of that age.