If it yields slightly to thumb pressure, then pick and eat it. Once it is separated from the tree, the clock begins ticking. They're really only good for 48 hrs after picking. After that, they get overripe and nasty (at least, that's my opinion). It is possible to refrigerate them and keep them for up to a week, but I don't like them that way.
If it is still solid green and hard as a rock, then you might have picked it too early (not ripe), in which case it may never taste good.
The thumb test is key. Sometimes pawpaws have a yellow, brown, black or off-greenish cast when they're ripe, but the color thing is not always reliable.