Agree with Bjorn, start with the cheap scenario first. Have you had the car for awhile?
Alternator test in garage= pull the positive cable from the battery while running, your car should stay running. If it shuts down, the ignition and accessories are draining the battery, meaning the alternator has no output.
Starter could be going, you mention hearing no click nothing. Bendix in start could be sticking from heat. Tap on it with a hammer, if your hear it click it will then turn over with the key. TAP not hit..
Could be hot start issue. Meaning vapor lock, fuel in carb too hot. But usually the car still turns over when this is the case, it sounds like weak battery or timing advance too far. Carb spacer will solve this.
