AFter I use the toilet, it flushes fine, but to get the tank to fill back up with water I have to take the lid off the tank and then either push the float down by hand or flip the little switch that is attached to it. I have tried cleaning everything (the rod, float, etc), but that has not help.

You need to adjust the float so it does not keep cutting off the water supply. The float is not high enough. Try that.

Twink is correct. Which fill valve do you have and we will help adjust the float. https://www.askmehelpdesk.com/attachments/plumbing/12875d1224099054-toilet-noise-clip_image001.jpg

Tell us which ballcock is yours. Regards, Tom