If you're only taking two minutes on your pushups, you aren't doing enough.
You want to train endurance and reach hypotrophy, so do alot.
First take the maximum amount of pushups you can do and break it down, either halfing it or lowering it a bit. (From 60 to 30)
Now start with a hundred broken up into sets (so 5 sets of 20 to start then two 50 and so on), take a max of four minutes per a hundred.
If you're struggling to finish a pushup?
Good keep going that's the point, and once you get used to that keep on adding more and more pushups and make your sets bigger. So from five sets of twenty and two hundred total to five hundred and two sets of fifty.
This helps you build both endurance and reach hypotrophy, both of which you need to tone up or bulk up.
Oh and RUN... cardio is good for you and your core and won't fuck you up if you do it right. Just eat a little more and track your macros (caloric intake) and micros (fat intake, protein intake) while cutting out empty calories and added sugars. Once you stop chugging soda and eating like shit you lose the taste foe things like that. It's hard at first but your body will thank you and it means you can fill up more on shit that actually has something in it. Instead of a milkshake, another serving of steak.
There are a billion resources online to caculate your needed fat and caloric intake, use them and work out and you'll be good and once you have a firm foundation and diet, joining a gym and using it to it's fullest extent becomes much eaiser. There is no where to go but up, don't start doing this tommorow.. do it right now. Start doing pushups and crunches and planks RIGHT NOW