Actually, two things come to mind. The first is the flapper. Your flapper should stay up till the contents of the tank are emptied. Some flappers are low flow flappers, and have an extra hole in them to cause them to shut sooner. Try replacing the flapper with a type that floats longer. Korky makes a good one. As far as refilling the bowl, it sounds like your fill valve (ball cock) is not opperatin properly. Typically they go bad after a few years and you can either rebuild them or replace them. I would suggest replacement, so much easier. But replace the supply tube at the same time. Make sure the tube that runs from the top of the fill valve is placed properly inside the overflow tube. Its what refills the bowl as the tank is filling after the flapper has shut. Without the proper amount of water in the bowl, the next flush will be whimpy.