The title is a bit misleading, since your ideal choice of exercises will always be dependent on your goals and abilities. The contents of this post are geared towards beginners and people with somewhat general goals that pretty much everyone will have to some extent. Read on to find out which movements I deem most important and especially why.
Although there are countless squat variations, all with their uses and limitations, I try to let everyone do at least some sort of squat movement. Besides overall strength and hip/leg strength, a proper squat at good depth can go a long way in preventing knee- and back issues by reinforcing correct movement patterns and maintaining hip mobility.
The deadlift is the most obvious example, but deadlift variations or even heavy kettlebell swings work wonders as well, for much the same reason as the squat. The difference being that the squat has more leg emphasis and the deadlift has more back emphasis (and slightly more hip emphasis). It also aids in teaching people proper mechanics to lift things in daily life, to maintain a proper posture and to learn how to use their trunk muscles to protect their spine.
The standing overhead press, military press and their variations are known for building upper body strength (more specifically: the shoulders), but they also aid in doing this properly when standing, thus requiring you to actively use your hips, abs and other trunk muscles. Another important point is that proper overhead pressing movements will maintain both your mobility and strength around the shoulder girdle, essential for shoulder health.
I always try to have the above three movements incorporated into every training regimen that I write for people. There are a few other exercises and movements that I usually have people do though. Not counting correctives or mobility exercises, I usually try to let people do some of the following as well.
Upper body pulls
Rows and pull ups add to overall back strength and may serve to reinforce proper posture and maintain shoulder (blade) health.
Bench presses, push ups and all their variations. They let you lift a big amount of weight with your upper body and thus serve little else than building mass and upper body strength. I rarely make this movement a priority, but it usually fits nicely with most people’s goals (that often revolve around building muscle mass and gaining strength anyway).
Power cleans or box jumps
Explosive movements are great for a variety of reasons. Not everyone will have the same need to learn these, but lifting heavy stuff onto your shoulders and being able to walk around with it is always a good skill to have in daily life. For that reason, power cleans are a better choice, but box jumps also help to make your back/abs/hips stronger. Power cleans can also be scaled more easily by adding weight. Box jumps, however, are a lot easier to learn.
Judo rolls, fall breaking, or even some fun exercises like spiderman walks. I’ve even had older people just lie down on a mat as fast as possible and come back up for 2 minutes straight. Harder than it sounds. The idea came from Dan John, who made a good point when he mentioned that not having any strength on the floor can be dangerous for old people. Being a bit limber and strong on the floor, easily getting up, breaking a fall… It sounds so easy when you’re young, but most people forget to maintain these simple but overlooked skills as they grow older.
There’s countless other exercises to pick from of course, but since everyone has at least some benefit from getting stronger and healthier, I always work with a solid foundation of the aforementioned movements. Isolation exercises, assistance exercises and other stuff are fine, provided that they build on said foundation.