I do a fair amount; what type of bike depends on what and where I'm going. When I'm taking a road route, I have generally just ridden a fairly normal road/touring bike, with a rear rack and panniers to haul my stuff. I generally don't bring a lot of food prep stuff, I just ride to where the food is. I bring a sleeping bag, lately a hammock to sleep in, sometimes a tent, a couple of changes of clothes, and some basic tools/spares for the bike. I've ridden down the length of Florida this way a few times, along with many short trips.
I've done a couple of off-road trips now and I'm working toward some much longer ones (Tour Divide, ~2700 miles down the Continental Divide). I'm riding a fat bike off road quite a bit; I'm still waffling over putting a rear rack on it and using panniers, or using the giant seat bags many people are using nowadays. The rack isn't suitable for very bumpy stuff or singletrack. I did one 250 miler off-road using only what I could carry in a backpack; that sucked, frankly. :-)
My advice is to keep doing what you've done already: take short trips and work your way up. Your gear selection and routes will evolve from there. If you haven't seen the Bob trailers, take a look. If I were hauling a family's worth of gear, I'd pick one of these.