1989 accord lxi starting problems

I was told it was probably the main relay but now I'm not sure. If I go somewhere and I turn the car off and then come back out a little later it wouldn't start for about fifteen minutes after I tried to start it. But sometimes it happens after I drive to work and let the car sit for eight hours and then I come out and try to start it and it won't, then I have to wait for 5 - 15 minutes then it will. Could it be the fuel pump? Or is it the main relay?

Or could be the ignition coil in the distributor

how would I tell if the coil is bad?

No spark and everything else checks out OK. There are greater possibilities of problems in the engine compartment then elsewhere because of the high heat of the engine. Main relays and ECU are a favorite target of do it yourself repair types but most problems are closer to the engine.