My guess is a loose handle or handle arm assembly. When you flush, the assembly is not being constrained appropriately and twists, meaning it can't pull up on the ball cock/flapper with correct force. That or something is up with the chain.
First thing to look for is if the handle is sitting properly. The face of the handle should be parallel to the face of the tank. It should have a solid fit in the talk hole with a small amount of up and down jiggle. It should not "twist" in and out more than a little if you pull out/push in on top of the handle.
Take the top off the tank. It will just lift off. Place it on a towel on the floor. This is not a "requirement", per se, but it keeps you from doing something stupid like balancing it on a vanity/seat and having it fall and crack. (NOTE: I do stupid stuff all the time).
There will be a lock nut on the inside of the tank for the handle where it passes through the tank wall. If it's not there, look for it along the arm and/or chain. Make sure the nut is installed properly and snug (doesn't need to be tightened hard, just hand snug).
EDIT: Unlike the water in the bowl, the water in the tank is clean water, having not been yet exposed to anything unsanitary. If you have to reach in to grab something that is in the water, you can then simply dry your hands/forearm afterward.