Congrats on the PR @e34bb098!!!
I don't do iron at all, but 3x5 reps seems a touch low for progression. Is it different than bodyweight training?
Well, I'm hardly an expert, but from what I understand, high reps are considered better for increasing size, and lower reps are better for increasing strength. E.g. a lot of bodybuilders use 3x10s on various muscle groups. But of course a weight that you struggle to hit 3x5 with, is impossible to do 3x10 with.
Re: bodyweight, yeah, it's different. I used to do only bodyweight and got into pretty good shape doing it (but I was also doing martial arts 4x/week and riding my bike everywhere). The issue there is that you will develop endurance but strength is incidental. Muscles don't adapt unless you force them out of their comfort zone, and they adapt to the stress they get. If you stress them with progressively heavier weights, they get stronger. If you stress them by making them progressively more tired, they get more endurance. (This is the "fast twitch" vs. "slow twitch" muscle fibers.)
Regarding progression generally, 3 sets of 5 reps is pretty well established as the best way for newbies to get strong as fast as possible. 5x5 is well-respected too (the legendary strength coach Bill Starr used this in football programs across the country for years). The reason is that when you are untrained, your body adapts very quickly. You're very far away from your theoretical maximum potential (based on your age and genetics). I have been adding 5lbs to each exercise for a while -- on each exercise I started with only the bar, then added 5lbs the next time, etc.
But eventually you stop making n00b gains (as they call it) and your progression slows way down. You are now much closer to your maximum potential and it takes progressively more work, with diminishing returns, to force another 1% of improvement. Instead of being able to add 5lbs every exercise, you will slow down to only being able to add 5lbs/week, then eventually 5lbs/month. Then it gets really hard, but that is the domain of elite strength athletes, which I am not interested in being. I am pretty close to the end of my n00b gains in squats and OHP, getting close in bench, and have a ways to go in deadlift and row. This is based mostly on how hard each on feels, plus the fact that I bounced off 295 pretty hard.
You also have to trick your muscles. Instead of doing the same routine, you might have to do a medium-light-heavy split. E.g., for my squat, I might do 3x5 @ 80% of my max on Monday, then 3x5 @ 60% on Wednesday, then try to do 3x5 on Friday and increase my max by 5 lbs. It gets complicated and there's a million theories about how to best program a routine that will take you beyond novice and into the intermediate range.
TL;DR: 3x5 is more than enough for progression if you are a beginner. (And thank you for the congrats.) :)