Your posts are pretty vague, but I'm guessing it's because you don't know much about this. Nothing wrong with that at your level of experience, but knowing more would help us make recommendations better suited to you. My first recommendation is to get into doubles and get proficient in them, then take your cavern class in doubles. Why train in a single tank if you plan on being in doubles?
Take a look at sidemount, also. There's no point in spending a ton of money on backmount gear to only end up buying sidemount gear a year or two later...lots of people have done this, me included.
Bigger tanks aren't always better. I can get just as far in a cave in my 85s as I can in my 108s. The 108s are huge tanks and my air consumption goes up when I have to swim them. Sometimes less is better. If you already have 130s, then by all means use them, but if you need to buy tanks, do some research. The right tank for you will depend on your height and build.
As for lift, you need enough to compensate for you and your gear. Unless you plan on diving dry, you'll probably need more lift for those 130s. I dove in Mexico a few years ago and got a set of 104s (heavy tanks). I could barely stay neutral with those tanks in a wet suit and a 70 lb lift wing. Also, if you plan on diving wet, get a dual bladder wing. You'll want the redundancy.
Tell us more about your future diving plans and we can provide more advice. Another thing you might want to consider is completing a course like Intro to Tech, Twinset Diver, or Sidemount Diver. You'll learn more about the gear and be better educated before you spend your money. During the courses I teach, I have my students try at least a couple of different tank sizes so they can feel the difference.