Until you can drop and do 75 pushups, do a lot of pullups, and do a few hundred situps, you shouldn't go near a weight set, unless you are worried about looking pretty and not practical strength.
Horsepuckey.
Weights allow you to build strength and endurance at any level, because you can control them. They also allow the targeting of muscle groups like rotator cuffs that get trashed by a "just do it" mentality.
I'm not talking about bench press, which I've become convinced isn't worth doing because it can trash the shoulders and biceps tendons. Dumbbell chest press, though, can work great, and allows the the vast majority of even fit people, who can't do 75 pushups, to get a workout.
But properly used, with education, weights and cables can allow a lot of functional gains. They also allow enough variety in how each muscle group is worked, to avoid repetitive motion injuries that are inevitable if one has a really limited number of exercises they do often.
The whole "do pushups and pullups" thing may be fine -- for a while -- if you're already fit and in your early 20s with no previous injuries, though it misses a lot of your body. Otherwise, though, jumping into boot camp calisthenics without much education about the body can
wreck you real quick.
Like anything else: knowledge is power. Don't just throw yourself into a sport or exercise, especially after a few years without it. It's easy to get caught in a loop of pushing yourself, getting hurt, getting better, pushing yourself, getting hurt, until some of the injuries don't heal. It's not the big injuries that are the problem; injuries to joints and small muscles, especially repetitive motion injuries, can be crippling.
Been there, done that. Ultimately got my CSCS.