It varies from species to species. I tend to rear in well ventilated boxes providing leaves from food plants. If doing it that way you need to refresh the foodplant every day or so. Alternatively you can have a growing plant (or at least a cutting in water) which means less frequent replacements (but make sure to block up any water pots or the caterpillars may drown themselves).

Line the bottom of the container with kitchen paper or similar to catch frass (a surprising amount) and clean out frequently.

When the things are looking to pupate provide them somewhere to bury themselves (if they are a burying species). Various options but perhaps the easiest is just to get some soil, microwave it to kill off everything in it, let it cool and use that.

Of course a significant proportion of caterpillars taken from the wild are infected with something or parasitised already and die despite your best efforts. If a parasite emerges that is worth retaining as they are seriously under-recorded.