Hey Joseph!
Sounds like you've got the right mindset - wanting to be healthy and to not rush the process. Good for you.
First of all let me state for the record that I'm not a professional or an expert here. I am one of those people who compulsively researches everything (*grin*) so I've done a LOT of research and reading in the area of weight loss, fitness, and health. That and I've lost about 80 lbs and kept it off for 18 months now. Those are my only qualifications.
Whatever I say here is based on what I have researched and what has worked for me. Now that said, I have a few thoughts for you.
The biggest part of being healthy and maintaining a healthy weight is EATING RIGHT.
When I first started wanting to lose weight and be healthy, one of the things that I really focused on was learning to eat REAL food. That meant cutting out packaged and processed foods as much as possible. Not just cutting out "junk" foods, but cutting out things that contain chemicals and preservatives and artificial ingredients, even if they were supposedly "good" for me or were low cal or low fat or whatever. One of the things I challenged myself to do was to eat foods that my grandparents would have been able to purchase and cook.
So for example, I don't eat "diet" food. I don't eat anything that comes out of a box or can that says "with added vitamins and nutrients". I don't eat anything that is specifically designed to replace nutrients that I can get from REAL food. I do eat fruit and veggies, oats, beans, pasta, whole grain breads, natural peanut butter, eggs and dairy, lean proteins (chicken, fish, pork, beef - pretty much in that order), and so forth.
Now I know that at 16 it's harder to make those kinds of dietary changes when you're not the one buying the groceries or cooking the meals, but if you are in a situation where you can ask your parents to maybe buy some things for you to eat/snack on over the summer - or maybe if you can spend part of the summer learning to cook and maybe doing some meals for the family - you might be able to tweak your diet some. One thing is that over the summer I eat a LOT of grilled food so maybe talking your parents into letting you learn to grill would help out some? I don't know what your home situation is, so let me know if my suggestions are way off base here.
As I said, what you eat is the most important thing.
When you're eating the right thing, it makes the exercise you do more effective - and it makes you able to exercise better (faster, stronger, etc.).
So from there, there are a ton of things you can do at home without a lot of equipment in order to build strength and maintain your muscles while you're losing fat. Basic bodyweight exercises are (IMO) the best way to start for someone who is out of shape and overweight. Things like squats, lunges, stepups, pushups, chinups or pullups, dips, planks ... all of those are great full body exercises that will get you on the right track. Mix in those items with some kind of cardio in short bursts, and you'll be on a good track to losing weight and getting in shape.
If you have the money to purchase some equipment to start off with, I'd recommend a good set of barbells with adjustable weight ranges. I know Target has a couple of sets that run around $35 (This is a good set here that will give you room to 'grow' for a minimal investment: )
The nice thing about a set of barbell weights like this is that as you get used to the body-resistance exercises above, you can start adding weight using this set. For example, I started out with body-weight lunges, and now I do them with a 15 lb barbell in each hand to raise the intensity. Same with stepups.
Another thing you can look into getting to help with the cardio is a jump rope. It's a simple thing, but 15 minutes of jumping rope will kick your ass ..
and it's a great cardio workout. You can vary the intensity and the speed to keep mixing things up and keep them from getting boring. And I think I got my jump rope from Target for $6.
Finally, if you have a little extra money, I'd recommend one of these for about $30: . I got one for my birthday this year and I love it. I've always wanted to be able to do a pullup or a chinup ... I think for me being able to do one is the thing that totally dispells the "weak girl" myth for me. I still struggle with them, but someday, I'll be able to pop out a dozen pullups w/out breaking a sweat! I'm determined!
I hope that helps you some and please feel free to ask any questions. Like I said, I'm not a professional, but I've been there/done that, so I'm happy to share my experience and research when I can.