One of the major mistakes I see people make that derails progress massively is putting too much time and energy into the wrong exercises. Everything I'm going to here is a major exercise that if it goes up over time, so will your strength and your physique. There is still a place for other stuff but keep in mind that they are accessories to the main lifts. Majoring in minors won't get you very far. Now, if you enjoy doing exercises outside of what I list here then by all means have at it. This article is written from the point of view of what will give you the biggest bang for your buck for strength and size.
1. The Squat
The squat is one of the most important exercises you can do. It is arguably the best exercise for developing size and strength in the lower body.
Both back and front squats are great choices. The safety squat bar is also handy if you've got shoulder issues. If you are a beginner, starting with a goblet squat can be a great option before you advance to the barbell. Squatting has also been shown to have a great carryover to many other activities such as sprinting and jumping, making it a cornerstone of any good strength and conditioning or performance focused programme.
2. The Deadlift
The deadlift may be one of the oldest and greatest tests of strength in human history. As far back as the viking and celts, humans have understood that being the best at picking something heavy off the ground makes you a strong S.O.B .
The deadlift does an amazing job of building the posterior chain muscles (glutes, hamstrings, lower back). Though these are the main muscles used, make no mistake that for a heavy deadlift almost every muscle is worked to some degree. I'm a fan of the straight bar conventional deadlift, IF the athlete can perform it safely. But whether you're using a trap bar, straight bar, or just a heavy rock, make sure your programme includes lifting something off the ground.
3. The Press
Oddly, this exercise is often overlooked in strength programmes. Many believe that pressing heavy overhead is bad for the shoulders. This is nonsense and quite the opposite is true. Learning to press heavy overhead with good form is key for building a strong and mobile shoulder complex.
The press is not only great for building highly functional upper body strength, but it is also a full body exercise. If you can strict press your bodyweight over your head then you also have seriously strong abs and lower back.
It is also has some of the best transfer of any upper body lift. As an example, I've seen many weightlifters with great presses who never bench, yet can bench impressive weights. However the inverse is not always true.
4. The Bench
It's become fashionable to bash the bench press and label it as 'dangerous' and non-functional. Look, I'm not saying that the bench is the only exercise you should do for your upper body. However anyone I've come across who can bench close to 1.5x their bodyweight or more is strong, period. And yes I've seen that strength translate to plenty of other push exercises.
Can the bench be dangerous for your shoulders? If you use bad technique then yes. And the majority of people butcher this exercise so it's no wonder it's gotten such a bad rap. Invest in your joint health and long term progress by finding a competent coach to teach you correct technique. I keep my clients to push-ups as their main upper body strength exercise until they can do 20 solid reps. Then we do weighted push-ups, and then dumbbell bench when push-ups become too hard to load. This means that by the time they do the barbell bench, they already have good upper body strength and coordination needed to perform the exercise correctly.
You don't have to bench, but you will find that most other horizontal pressing exercises like push-ups will become too hard to load safely once you reach a certain strength level. For all it's supposed dangers, the bench is actually very safe and stable to load, and can be progressed for years on end. Just don't be an idiot with your form. If you want to get the basics of how to bench, check out this video I made.
5. The Row
All of the big lifts already mentioned require an immensely strong upper back. Rows do a great job of achieving this, along with keeping your shoulders healthy.
There are a lot of different rowing exercises that work well. My personal favourites are the Pendlay Row and dumbbell row, although they require decent mobility so do NOT do them with a rounded back. Instead try the chest supported row or ring row.
6. Chins / Pull - Ups
Much like the row, chins and pull-ups are great for building the back, although now in a vertical pulling motion, which will work some different muscles and improve your overhead shoulder mobility.
This doesn't have to be complicated; just do some kind of chin-up or pull-up at least twice per week. Don't worry about anything outside of bodyweight until you can do at least 10 perfectly strict reps. That means starting from arms fully extended and at minimum getting your chin over the bar each rep (chest to bar is even better). Use no leg swing or momentum.
If you claim to be able to do 20 strict pull-ups you are either
1) a liar, 2) very skinny, or 3) very strong.. and it's almost always number 1 or 2.
Now, am I saying that these are the only exercises you should do? No. What I'm saying is that these are the only exercises you need to be concerned about training (i.e. getting progressively stronger at).
Nobody cares if you put 20 kilos on your bicep curl, and doing so won't make your arms as big as you think. Exercises targeting your arms, abs, glutes etc. are important to some degree, but they shouldn't be your sole focus. These are what I would call 'JGID', or 'just get it done' exercises. They need to be kept floating in the background for health and muscle mass, but in the grand scheme they are a tiny piece of the pie. Conclusion
Put the bulk of your training focus into getting progressively stronger on some version of the 6 compound lifts mentioned above. They will get you 90%+ of the way to where you want to be.
Sprinkle in some smaller exercises at the end of your sessions to target abs, hamstrings/glutes/low back, arms, rotator cuff, grip, and of course cardio. Just keep them in perspective as small pieces of the pie.