Welcome to the forum. I'm sure you will get lots of answers. We camped in tents when our kids were young except when we used my dad's motorhomes (he had several over the years). We went from tents to a hybrid when my knees got bad enough that I didn't want to sleep on the ground. We moved from the hybrid to our TT when we wanted to stay longer and wanted to be more comfortable.
The amount to budget is very open. If you camp in a national forest or boondock then the cost of fuel and your food is basically it. If you stay fairly close to home and cook in camp like you cook at home the cost difference is negligible (if you don't have monthly trailer payments :) ). If you stay in state parks the cost is fairly low. In NM the rates vary from free to $18 depending on the amenities. If you camp at an RV resort you could spend over $50 per night. So the cost varies depending on what you want. Sorry, I can't help with a bunkhouse trailer as I have never owned one but I'm sure many will chime in. At any rate, camping is a great way to get closer with your kids and I know you will enjoy it.