Say you have K different outcome types that are independent in the sense that only one of them can happen on any given roll (e.g. your example of 1,2,3, or 4-6).

Let p be the probability of any of them happening on a given roll. Then p is just the sum of the individual probabilities. Use the term 'event' to denote any of the outcomes happening. So on any given roll, you have probability p of an 'event' happening, and when it does happen, it corresponds to one of the outcomes.

So we have decomposed the problem into the following two subproblems:

1) What is the expected number of events that need to happen until you have hit all the outcomes? I think this is the coupon collection problem? https://en.wikipedia.org/wiki/Coupon_collector%27s_problem

2) How many rolls do you need in order to get the above number of events (presumably this is negative binomial, since the probability of an event happening is independent on each roll and has probability p)