Assuming your aim is to get better at martial arts not bodybuilding (hence strength training: lets take Rippetoe's 3x5). Which would improve performance? Also can they be done side by side? because I know that after lifting heavy you have to wait a day and train on alternate days.
I had a lot of success doing bodyweight training for the couple of years that I was more-or-less forced to, living in a developing nation with no real gym options.
If you go the bodyweight route, I'd recommend these strategies:
- Realize that "some pushups and pullups" isn't going to cut it. Just like most people in a gym have no idea what they're doing, most people doing bodyweight training are wasting time.
- You need to effectively target your major muscle groups, maintain a balance of development, and progress through more and more challenging movements.
- A lot of people view bodyweight training as more simplistic then weights, but in reality they both have similar challenges: maximizing (in your case) power and force production throughout your body in an effective way.
- Bodyweight training, for me, includes such things as gymnastic rings and plyometrics. The increased power work doing plyometrics and the increased leverage with rings really is a game changer over just you and your body. No matter how you slice it, hamstring and back development is poor at best with bodyweight training alone.
- This is one of my preferred bodyweight books. For rings, and really other great sources of gym-less activities, beastskills.com is regarded by pretty much any athlete as a terrific resource.
If you go the barbell route, I don't see any reason why you shouldn't start something like Starting Strength's novice program. If you're past that, the Texas method is a template you can eventually modify, and it was designed for Olympic lifters originally.
I'd say to jump right into Olympic lifts, but the background months of properly performing big barbell movements is really necessary.
Either way, you're going to want to aim for power and strength. Olympic lifts, box jumps, shoulder stability, pistol squats with a jump at the end, etc. There's a lot of range in there to pick from, but whichever way you go make sure you're targeting power.
Knuckle pushups, bodyweight squats, and sit-ups, as a program, would mostly increase muscular endurance and cardio. A 3x5 program like Rippetoe's Starting Strength would actually increase your strength and power.
Muscular endurance is great for fighting, but A) you're probably already doing those exercises in class and B) if you're stronger you have better muscular endurance anyway. So rational general strength and power training is better for martial arts than bodyweight work chosen because it's bodyweight.
You can lift alongside martial arts training, but you have to take care for lifting not to interfere with skill development. Generally this just means 1) scheduling your lifting so you're not beat down when you go to class, and 2) adding weight less frequently than recommended for someone not playing a sport. With a T/W/Th/Sat karate schedule, I'd definitely lift either after class on Saturday, or on Sunday. If I managed to lift on Saturday then I'd try to lift on Monday too. If I were feeling good about my training, not feeling overworked or under-recovered, then I wouldn't mind lifting Friday and Monday (but not Saturday or Sunday). I'd add 2.5 or 5 pounds to each exercise every two or three workout sessions.
If I were feeling really good about training, full of energy even after lifting twice a week and doing karate 4x/week for a few months, I'd try adding a third lifting session. I wouldn't really care where it went.
If you split your routine up (chest one day, back another, biceps another day) etc, then you don't have to gym on alternating days, you could train daily.
You won't necessarily bulk huge by going to gym but you will get that strength and core that you won't by just doing pushups and exercises that use your own body weight. It just won't happen, your stabilising muscles won't develop as much as you aren't putting as much strain on your body.
People are often afraid to join a gym because they don't want to look like a bodybuilder but those guys dedicate years and thousands into the right diets to look like that, it doesnt happen easily.