I will do my best to answer your questions in the order they were presented:
Question #1: "What kind of training should I start with?"
Answer Part A: The first thing you should do is gradually get yourself into a workout program with an aerobic element. If you start into a training program too fast, you will likely burn out and lose interest. Start off slow. Aerobic exercises will increase your metabolism, burning fat - and combined with a reasonable diet, excess pounds should begin to disappear. Weight training can be introduced slowly as well and will provide great results when used in combination with aerobic exercises. Again, weight training should start with low weights (high reps) first, and then gradually increase the weight and lower the reps.
For tennis, I would highly recommend focusing a weight training regimen on your core (stomach/torso) and legs (quads, hamstrings, calves). Power is generated from legs, not arms or shoulders. Always remember to stretch thoroughly both before and after your excercising. Hamstrings and calves are prone to tears and cramping if not developed slowly and with poor stretching habits. Always stay hydrated with plenty of fluids and include a source of potassium in your diet.
Answer Part B: As far as tennis training goes, I would suggest finding a USTA Certified Pro in your area and try to get some lessons in once or twice a week depending on your budget and available time (you sound pretty busy). USTA Pros are all over, and can be found either in the phone book or through a network listing at www.usta.com. Your local park and recreation department may be able to help you too.
Don't feel obligated to stay with one coach if you don't feel comfortable working with him or her. Try to find someone who has experience with someone of your level. It may take a few tries with different coaches before you find someone you feel comfortable with.
Once you get the basics down pretty well, and want to progress to the "next level," I would very highly recommend attending one of the Bolletieri (or similar) tennis camps that are available in all regions of the U.S. They typically run in single week sessions and amount to about 9 hours of tennis per day. It's expensive, it's intense, but the benefits of attending one are tremendous for new or already semi-competitive players. This is where you will learn to be a competitive player in a relatively short amount of time.
Question #2: " What types of abilities and physical assets should a tennis player have?"
Many people here on Yahoo! Answers may have their own opinions, but from experience, I feel the following are the most essential for a semi-competitive (or higher) player: Good hand/eye coordination, good anticipation, stamina, strong legs (for movement and quickness), a strong core, mental toughness, and an overall well-conditioned body (conditioning can vary depending on the level of competition).
Question #3: "And what are my chances of ever playing competitvely?"
Just like with anything else, the more you invest of yourself, the more you will get out of it. If you casually get into tennis, it may take years and years before you would see any sort of success in local tournament competition. However, if you really dedicate yourself to the sport, find a good coach that can help you with your game - your skills will improve tremendously in a shorter amount of time.
Now, your definition of "competition" may be different than mine. If you are referring to collegiate competition, depending on the Division ranking of the school - you might not be at the level you need to be when enrollment starts. If you would like to compete in local tournaments and USTA events, that might be more feasible in a few years after some solid training.
I've personally seen (and trained) some people who went from picking up a racquet for the first time to being captain of a high school tennis team in four years (4.0 - 4.5 level). Granted, they had good coaching at the start and they dedicated themselves to the sport practicing every day and seeking the instruction from a high-end tennis camps (in this case Bolletieri's).
Question #4: "Wouldn't colleges rather recruit high school players--not rookie adults?"
Division A colleges likely recruit from high school, but if a walk-on does meet their standards, they will likely be accepted too. Division III schools usually hold try-outs where a coach would see for the first time the level of talent trying out for the team.
Question #5: "Are there any teams or programs for adults at any level?"
Yes! Look no further than the USTA. They have numerous programs for adults at most any level and they are held all over the U.S.- - and tennis is much more fun when you participate with people around your own ability level.
To sum this up, you are 19 and young. Don't get discouraged by what others might say about you being "too old" or whatever. 19 is young, and tennis is a sport that you can play for the next 50 years of your life. If you decide that tennis is what you want to do, and you put forth the effort and discipline required to meet your goals, you WILL achieve them. Depending on the resources available to you and your level of committment, you could conceivably be a 3.5-4.0 player in about 4 - 5 years. At that level, you should be able to enjoy at least some minor/local tennis events in your area. Everybody's development is different, so it is difficult to predict your rate of improvement over time. Your reference to wanting the opportunity of playing advanced players makes me think that you would want to hold a USTA rating of 5.0 or higher. Again - - practice, discipline and dedication will determine how quickly you progress.
A goal isn't a goal unless you write it down and place it where you can see it every day. Make small goals that act as stepping stones to the ideal place you want to be. This way, you can measure your progress, and have something to feel good about when you accomplish them.
I hope this helps. If you have any other questions, feel free to contact me directly.