The “Without becoming a fatass” part will be completely dependent upon nutrition. So:
To gain size and strength, you obviously need to be in a caloric surplus. Not a huge one, but if you’re trying to gain size, you can’t expect to make any kind of noticeable progress on maintenance calories. It doesn’t take a big caloric surplus to gain size properly, it’s largely dependent on the individual, their training, genetics, metabolism and numerous other factors, which is why it’s important to take advice from someone who’s been there, done that, and has the personal experience to pull from, rather than let’s say, a potential keyboard warrior who does not have said experience and thinks a guy who walks around at 5’6" and 230 is a natural.
If you want to gain size, it’s not just about nutrition, you need to train largely in the “hypertrophy” zone to gain muscle mass, which means more time under tension. Compound and isolation movements should be utilized. I’d highly recommend you give this article a read which talks about the difference in training methodologies between bodybuilders and powerlifters, and why bodybuilders look to have more muscle.
To gain strength, some lower rep and strength based work is needed. Maybe starting out the workouts with 5x5, or something similar, with of course squats, dead lift variations and compound movements incorporated.
I’d recommend a standard bodybuilding template, like this one:
With the first exercises being more in the strength focus. So if you’re starting chest day, maybe do heavy incline barbell or dumbbell presses for 5x5, then move on to 8-10 and 10-12 rep ranges for the rest of the workout. Gaining size isn’t just about rep ranges, it’s about knowing HOW to train, establishing a great MMC (mind muscle connection) with slow controlled negatives, initiating every movement with the working muscle and keeping tension on the working muscle throughout the entire concentric and eccentric phase of every rep.
To ensure one doesn’t become a fatass, I’ll repeat the same thing I posted earlier. Find your TDEE/maintenance. If you’re not sure what it is, use a TDEE calculator, plug in the info, and eat at maintenance for a couple weeks to make sure it actually is your maintenance. Once you know, bump to 10-15% surplus. Protein should be ~1g per pound, fats 25-30% of total cals, and fill the rest with carbs. Depending on one’s metabolism and body type, some people can get away with more fast digesting carbs than others. Personally, I stick with slow digesting carbs, except for post workout, because I do not have a naturally fast metabolism. Eat consistently, keep an eye on the scale and mirror, and if the scale goes up and conditioning stays tight/acceptable, you’re good to keep going. If you start gaining more fat than you’d like, scale back slightly on cals/carbs, maybe add some low intensity cardio, lose a few pounds, and then go back to gaining.
These are just my opinions based on my experience, that I’ve found success with and so do my clients. I’m not saying there aren’t other ways to make progress.
I’ve made significant improvements over the years following these methods and they’ve helped me earn some overall wins and a pro card. But hey, I’m no doctor.