NAUI has been around for a very long time; PADI is young in comparison; both are very well known worldwide.
Cave diving: you are only qualified to do that after taking a qualifying Cave Diver Course through a different agency, such as ANDI, TDI, and others. Check out sites on WKPP, DIR, GUE, and others for more information on getting that cert. I implore you NOT to do a cave dive without proper training. Most training agencies that do cave diver training will take you after Open Water Diver courses, under the premise they want to get you before the "bad habits" set in.
120 Feet: well, the recognized recreational limit is 130 feet. However, with proper training and equipment, one can go much deeper on open circuit scuba. You may wish to contact a technically oriented agency to answer the going deeper questions you might have. Or, continue to post here and/or other discussion boards. You are bound to get lots of information that way. Again, check out TDI, ANDI, GUE, WKPP, and others to get a start.