Recently, but without any consistancy, I've run into a problem
starting the car ('93 Legend, ~145K miles). It started about 4 weeks
ago. The first couple of times, I would crank it, and it would try to
start, but not turn over. Pull the key, wait a minute and it was fine.
Didn't think a whole lot about it.

Recently, I've been doing this - put the key in, turn it to the second
position and wait for the check engine light to go out. Starts up fine
when that happens. If I wait a few seconds and it doesn't go out, it
won't start. It sounds like it is not getting any fuel. When this has
happened, the last two times I was able to get it to start by pulling
the key, removing the gas cap, putting it back in and turning to
second position, then the light would go out and it started. Happened
when I left work yesterday. The last time it happened to me prior to
this was when I left it in the lot with the windows closed and the
whole inside of the car was extremely hot. I left the windows closed
again today as well since it looked like rain. It sat in the garage
for about 90 minutes when I came home from work, and when I went to
start it (position two etc), it fired right up.

Looking around, I see all sorts of things - main relay, fuel pump,
ignition control module and so on. Should I look at replacing the main
relay first, fuel pump, gas cap or is there something else I should
look for?