You mentioned that you are cleaning up your diet. That's a great place to start. Eating right not only is great for you physiologically, but it mentally gets you in the right mindset. It's a win-win. This may sound dramatic, but I didn't realize how good I could feel until I starting eating right.
I'm not sure what you mean by "utter shit" (there are many flavors of utter shit), but when I was 15 I ate cereal, skim milk, peanut butter sandwiches, the occasional apple, pasta, and hot dogs/hamburgers. No vegetables, and no nutrition of any kind really to speak of, really just wheat, peanut butter, crappy milk, a little bit of straight-up junk food, and the occasional processed meat. My parents fed me a multivitamin, but those are basically feel-good do-nothing pills in the face of a shitty diet.
You don't have to go crazy, but cut out the obvious junk, sugar and/or grains, and focus on vegetables and meat/eggs. Protein is important, but not as important as the whole-chicken-for-lunch-chased-with-a-protein-shake crowd wants you to believe. You'll get enough in any reasonable amount of meat, whole milk (if you enjoy drinking milk) and eggs.
Learn to cook some basic things. Cooking doesn't have to be fancy or incredibly time-consuming. Scramble some eggs, when you get good at that, make an omelette or breakfast skillet. Practice chopping vegetables, cook them in a pan with some meat and you've got yourself a stir-fry. Make some burgers from scratch. Steam some vegetables. Bake a potato. Hell, cook a potato in the microwave. Roast a chicken. Look up slow cooker recipes if you have one. I didn't do any real cooking (beyond essentially boiling water for pasta) until I was 21 and out of my parents' house, but I wish I had started sooner.
With that dialed in, start simple on the exercise end. You mentioned push-ups and pull-ups as goals. I'd start with push-ups. I know there are different stages of "pre-pushups" that can help get you there. Then once you're able to do a few real push-ups you could buy a pull-up bar (they're cheap) and mix that into your routine. Then you can break into the world of barbells if you want, and I would heartily recommend that once you get to that point. Barbells are not just for bodybuilders; they are amazing for average schlubs like me too, and have all kinds of benefits beyond putting on muscle.