I want to invest in some hotel pans for home cooking, what size and quantities should I buy?

Hotel pans are primarily used in institutional settings because everything is already made to fit them. Steam tables, prep coolers, etc, are all made to fit hotel pans and their fractional sizes. While I agree with the comments that they aren't generally the best pan suited to many tasks, they are great because they are fairly inexpensive and can be stacked and nested so easily.

There are not, however, any specific sizes that you must have. Even restaurants only buy sizes that they need for specific tasks. For instance, I have worked very few places that had more than one or two full sized hotel pans, as those are usually only used as the water-holding layer of a steam table (or occasionally to do roasts and the like). Basically, get the sizes that you think you will use. For instance, if nothing bigger than a 1/2 pan will fit in your oven, don't get any full pans. If you primarily want them for refrigerated storage, only get the plastic cold table pans and lids in sizes that will fit in your refrigerator.

If you are primarily using them for storage, I'd highly recommend getting Cambro or Carlisle storage containers instead. They come in standardized sizes, have color coded lids, and nest neatly. They also have snug fitting lids, which hotel pans lack. The plastic cold table pans generally have loose lids that work in institutional settings for things that won't be stored longer than a few hours, but wouldn't be practical at home.