The engine will not crank but if you bypass the starter, it will turn over but not start?

We have had this car for about five months. About a month after we bought it, we were out in it, shut it off to go in a store and it would not start back, no crank or anything. We waited about a half hour and tried again, and it started the first try. This happened about three more times with it taking longer to start back until this last time it has not started in over a week. It will not crank but if you bypass the starter, it will turn over but not start. The electric fuel pump is not kicking on, so basically nothing on the motor is working but everything on the inside comes on. We put in a new battery and checked the fuses and relays.