I want to make sure I have a good grasp on all the practices of caring for a bonsai before I really have to start pruning.
Good idea, there will probably be very little need to do much in the way of pruning early on. Pruning very young material is often counterproductive. Once you receive the plant evaluate it and begin to think about what size and style the tree will eventually take.
With the 3" and 4" ones about how long do you think it will take before I should start pruning it?
It depends upon the goal you set for the material on hand. if you want something with a one inch trunk it makes little sense to constantly prune and mess about with it. If you decide to aim for a much smaller image you can begin the training process earlier.
And I want to get a pot for when I am ready to replant the bonsai, even if that is a long ways away. What size would you guys recommend for me for later on down the road?
There really is no need to rush but something a few inches larger than it is in now will allow for some root growth. Don't go too large, perhaps 6 inches for a plant currently in a 4 inch pot. There are also plastic pots that are good for this purpose, they are shaped like proper bonsai pots but don't cost as much. You will go through a series of these over the years until the plant approaches your design goal. Then is the time to begin to think about the final plant/pot combination that is bonsai.
And I'm realizing all I need to do for a humidity tray is get a tray at the garden section of a store and put rocks in it. Am I correct?
Correct, it need not be anything fancy but the larger the better. More square inches equals more surface area from which water will evaporate.
Do bonsai require anything special in terms of food? I've heard some people like to give their plants something like a nutrient supplement. Is that good for a new bonsai or not necessary?
Bonsai do require regular fertilization just like any other potted plant, they do not require any bonsai specific product or any exotic brand. More important than what you use is that you do fertilize regularly during the growing season.
Oh, and I was wondering about light for this bonsai. Should I get artificial light for it? Or do pomegranates not require quite as much lighting.
Probably, unless you have a really good southern exposure some supplemental lighting will help keep it healthy and thriving. This site has information on growing indoors including lighting.
Here is a care sheet for Punica: