Will not stay running

Okay, so if the car sits for more than four hours it has a hard time staying started. It is not to hard to get it running just for it to stay running. But after about thirty seconds of running it does okay most of the time. When it does not do okay it likes to shut off when we stop but we start back up with out a problem. When we first start to move the car is really sluggish and slow to respond to the gas pedal. Some extra details are when at a stop or idling it has a harder. Do not know how to put this but if the car had a heart beat every four or five beats is hard like you feel the car shake a little when it happens. It all so now if we have to come to a quicker stop, when we hit the gas there is a hard thump coming from what I think is the transmission. It started doing this soon after we took it to a dealership shop to have the radiator replaced. That is about as much details as I can give. Please help if you can it is my mother's car and I am sick of these repair shops ripping her off for things the car does not need.