2006 Toyota Solara Battery vs. Starter vs. Something else?

The past few days my car has trouble starting. It always did on the first try but something was wrong. Yesterday I went to start my car at 8:30 am and it did the same thing starting but my clock said 1:00. So I told my dad who looked at the battery and the connections were completely corroded. He disconnected the battery and cleaned both sides of the connection. Then when I went to start my car it was completely dead and would not start with a jump with the jump cables connected to the connection, only with them connected to the battery itself. I ran the car for about 20 minutes straight to try to refresh the battery and then let it sit over night. This morning when I went to start the car it was dead again. I got a jump, and this time tried connecting the jump cables directly to the battery and the car started immediately. I drove for 15 minutes and 8 hours later the car is dead again. The only noise I am getting is a clicking noise. My horn, power windows, all dashboard lights are working properly. When my car is on the battery warning light is not on. The green light on the battery itself is visible, not extremely bright, but my dad says bright enough. I dont want to replace the battery if there is something else wrong and and then drain another battery. But I dont understand how it could be the starter because how would my car start on a jump so easily. Also, if the connectors between the battery and the car are too corroded to work properly why would the jump cables connected to the battery work because doesnt the electricity still have to run back through those cables into the car? Any help/insight would be fantastic. Thank you.