A leaking flapper most likely causes both symptoms. This is the round seal in the bottom of the tank connected to the chain for the handle. When you depress the handle, the flapper pops up to allow the water in the tank to rush into the bowl and cause the toilet to flush.
The flapper is easily removed and replaced. A new flapper costs around $2-$5 at any hardware store and replacing it will likely fix the problem. Be sure to remove the old one and match it up to the new one to ensure proper fit.
Once the flapper is replaced, adjust the toilet ball (ballcock) gently downward to lower the water level in the tank. Remove any toilet bowl cleaners from obstructing the flapper in the tank.