Ditto, everybody above!
Do your battery cables have a green or white powdery corrosion? You said they do not look healthy.
The black battery cable is the main ground, and it attaches to the engine. Is this end of the battery cable clean and making good contact?
There is a ground wire attached to the back side of the intake manifold, and then to the firewall on the passanger side.
There is a smaller ground wire next to the battery, toward the rear and screwed into the fender well. The two attaching bolts for the crossover plennum (passanger side) are "grounds".
There are a few others, check the service manual for their locations.
Check your battery. A poor connection could result in poor charging, and therefor the weakness you describe when trying to turn over the engine, even though it is a new battery.
Check your charging system (alternater). If you are running off of the battery, and not recharging it, you would have the symptoms you describe, that is a weakness in trying to start the warm car. After the car has sat for awhile, the battery recovers to a degree.
Even though you have a new battery, it could be a bad one. It happens! The same is true for the starter. An amp draw test will tell you if the starter is OK, or not.
Radio Shack, Sears, and most auto parts stores sell a good muti-meter for testing electrical systems for amperes, volts, ohms, and continuity.
A good meter can be had for $30 to $50.
If you are going to do your own mechanical work on your car, you are going to need one. rangerj