It's the difference between "training" and "exercise". If you want to burn calories or look good at the beach, it doesn't matter what you do. That's exercise, which is what the majority of people do. If you actually want to do something and develop your skills, then your work is focused. That's training.
So say you are in lifting heavy weights like a bodybuilder? What are you training for? You're training to lift very heavy things a short distance for a few reps in a short period of time. Period. That's why bodybuilders frequently can't do calisthenics movements, or pass military fitness tests, or climb a rock wall, or even go for a run. How does that skill help you in a fight? Combat sports are endurance events, particularly boxing. At the professional level, you're basically doing the equivalent of sprints for short bursts, but you're doing it for maybe 48 minutes at championship level with short rest periods, and someone punching you in the head and body to boot.
So if you put on bulk, you're likely putting on primarily fast glycolytic fibers, which will fatigue fast, and give you extra bulk to move around for very little sustained power, which usually leads to slower strikes and slower feet, in addition to the hit on your gas tank.
There ARE ways to train strength for fighting, but you'll likely do a lot of that as part of your classes anyways. So don't worry about it. You'll get very fit if you're putting in multiple fight classes a day every day, with the occasional rest day.
As for other advice, focus on fundamentals. Even at the championship level, I see guys who don't have their fundamentals down, particularly footwork. If you have good footwork and timing, you will beat a lot of guys who have more skill elsewhere but don't have those aspects.
Positional fighting, for instance, is a lost art these days, but really gets to the essence of combat sports.
For instance, take the concept of the centerline. Power goes in a line between your hands and feet towards your opponent. It's very difficult to generate power on an opponent who is outside your feet. You can't land good strikes, and you can't take them down. If you understand positioning, you can put yourself in a spot within your power and outside theirs, even if you're in range to be hit. So coaches will tell you to circle away from the power hand, but if you do that within their feet and within range of the hook, you're going to get caught by the hook (or round kick). That's what happened to Joe Joyce in the rematch v Zhang. If he had stepped around out of range of the hook, but still in range of his cross, then it could have been a good strategy. But you've got a bigger distance to travel then, while they just have to pivot on the spot. So you need a shuffle step and to cover it with blinding offense. He didn't do that, and he got knocked out, because he doesn't understand positioning.
Point is, put in the time and have goals for yourself. Train smart and you can make progress faster than people who don't do the same.
What should I do to increase my strength and size as I am lean 121lbs at 5'9? Also, wouldn't doing a strength oriented program such as Starting strength or Stronglifts 5x5 be good for MMA as two fighters whose skills are equal, the strongest fighter of the two will win.