I am a 16 year old guy, about 5'7" (1.7m) and 232lbs (105kg). I have only started to go to gym for about a week now. Since the gym does not have a trainer, I have only done cardio (elliptical 20 min,cycling 10min and treadmil 20min).
Should I just continue doing this till I am at say 180-200 lbs (81-91kg) and then start doing weight training or should I start now?
At 1.7 m and 105 kg, your BMI is 36.3, which is above the 99th percentile for males your age and if you are experiencing any health conditions related to your size, this puts you in the morbidly obese category.
In general, anybody who is interested in embarking on a diet and exercise plan would do well to consult a physician before doing so. In your case, I would say it is a requirement, because the potential for injury is something that should not be ignored. It is not the role of this site (or any online forum, for that matter) to dispense medical advice, so I strongly urge you to exercise in a supervised setting (i.e., you should have a trainer).
That said, let us assume that you've got physician approval for whatever exercise and diet plan you are going to do, and you've got adequate supervision. Let's answer the specific question you asked: should you incorporate resistance training in your exercise plan?
The simple answer, of course--and this is true for just about anyone--is yes. Resistance training, specifically weight training, has numerous benefits that cardiovascular training alone does not target:
The unasked question that is implied is, "what kind of resistance training should I do?" And this is something that is difficult to answer, because every person is different. We each have different health conditions, physical needs, body shapes, ages, and strength goals. What works for me may not work for you, and vice versa. That's why a supervised plan is likely to be safest and most effective.
Up to now, I think I've largely stayed away from personal opinion, so what follows is only my thinking; others may disagree. In my view, you are already carrying at least 40 kg of excess weight with you, 24 hours a day, 7 days a week. If you do a bodyweight squat, that's like me doing a barbell squat with an extra 40 kg (well, not exactly, of course, since the weight is distributed differently). But the point is, your own excess weight is already providing a nontrivial amount of resistance on your muscles. So your initial resistance training should be focused on bodyweight exercises (e.g, push-ups, crunches, bodyweight squats, etc). You can of course do free weights or weight machines, but don't look at a plan that is designed for non-overweight individuals and think that that's where you need to start. The intensity of your resistance training should be extremely low initially: this minimizes risk of injury and overtraining and allows gradual adaptation.
The one major mistake I think that people make when starting regular exercise for the first time in order to lose weight, is to overdo it. You didn't gain that weight overnight. Why do you think you can lose it overnight? Cardiovascular training should be the central focus of your workout plan at this time. If you can supplement that with resistance training, fantastic. But do not sacrifice the former for the latter. For you, just walking is going to make you lose weight, provided you adjust your caloric intake. And it will happen fast. Not overnight, but a lot faster than you might think, if you are consistent. And the more weight you lose, the more you can shift your program toward weight training, because you're not carrying as much extra mass.
Start weight training as soon as you know how to do the movements. You can probably start fairly safely with things like body weight squats and push ups. I would get a trainer or someone who knows how to lift before I started things like back squats, bench press, overhead press, etc. Weight training, in addition to cardio and eating sensibly, will help you lose weight. The exercises themselves will burn calories and as they repair they will burn more calories.
You goal should be more focused on body composition vs numerical body weight. As such, there's no reason to not do resistance or weight training.
Phlegon's answer about starting with body weight resistance via what's generally considered calisthenics is a good one. That's also a way to do some of the exercises at home and build up some strength and fitness if you're at all self-conscious about the amount of weight you'd be pushing at the health club.
So.... doing resistance exercises doesn't have to mean developing bulk, if that's what you're worried about. Lower weight, higher repetition sets will develop some strength, but will develop more fitness, tone and endurance, which will also help with your general stamina for your other aerobic activities.
Variety is good, so why not do more types of exercise? It's better, mentally, to keep you from getting stale, and a cross section of exercises and types of exercises gives you good functional fitness, where the advantages and drawbacks of any particular exercising method even out with cross-training with other methods.
