I'm guessing, from your questions, that you have never done anything like this. If so, there's a lot to know, and a lot of gear to accumulate. You want a bike that is strong enough to carry a load, particularly the wheels. I think a dedicated touring bike is best. There are many to choose from. I also think mountain bikes are fine, particularly hardtails or rigids. You can successfully tour on just about any bike, but you'll have better luck with some than others. Avoid carbon fiber frames. Longer geometry is nice, and long chainstays are nice if you use panniers and have large feet, to avoid heel strike on your panniers.
You need lightweight camping gear - sleeping bag, pad, and shelter (a tent usually). I also suggest cooking gear - a stove, a mess kit, a plate, coffee mug, cutlery, etc. I met a guy riding across the country who didn't have the ability to cook. He said finding food every day was a huge hassle, and he wished he could cook. I think you need rain gear. I like rain gear suitable to ride in, because I don't want to get wet and cold while riding. You definitely need rain gear in camp, unless you're going to get a motel room every time it rains. You can't always predict it though. You might go to bed in nice weather and wake up in a deluge.
I prefer panniers front and rear, but I've also toured with only rear panniers (short trips) and a Bob trailer. If your bike isn't the best for carrying panniers, a Bob trailer can be an easy solution. However, again, I've heard you should not use a trailer with a carbon fiber frame.
I would not tour with anything heavier than a Camelbak on my back. I used to carry my books in a backpack when I rode to college every day. However, that was an 8 mile roundtrip, and even then it got heavy. If you're riding lots of high mileage days in a row, that weight will become unacceptable fast.
I did a lot of backpacking before I started bike touring. One of the attractive aspects of bike touring is that you usually don't have to carry much food, as you will be passing grocery stores along the way. You usually pass places that have water too, but not always. Particularly out west, there are long stretches where you might not be able to fill your bottles. One option, if you're going to be riding past rivers and lakes, would be to carry a lightweight filter. I wouldn't want to filter water too often, and the filter might be one more piece of junk to add to your pile of unnecessary things adding to your total weight.
Most touring bikes have three water bottle cages. That's usually enough for me. Last summer I was in some places where getting water might be difficult, and riding on many days where the temperature was over 100 degrees. I carried my three water bottles, plus a Camelbak. I never ran out of water, but there were some times when I was on my last bottle before I found a place to fill up. I learned to look at the map and plan my water accordingly. If I was going to be passing places with water every hour or two there was no need to fill all my bottles and Camelbak.
I suggest you keep reading this forum. Also get on CrazyGuyOnABike and start reading journals. There are also interesting articles and discussions there. Find out what other people do in terms of equipment, routes, etc. Think about what would work for you and start asking questions about details. Take a few mini-tours and see what works and what doesn't.
My feeling is that if you are adventurous and like to travel, there's nothing better than bike touring! Have a wonderful adventure!