Normally I’d recommend 4 exercises that target a push, pull, hinge and squat each trained in a rep range of 3-8, depending on age, injuries, goals etc…
However, when the goal is strength for normal daily function and quality of life, especially with an irritated/painful lower back, my thought process would be to pick exercises that support the movements you want to be strong at in daily life without irritating your lower back any further.
The movement pattern of the exercise doesn’t have to be exactly the same as the one you do in daily life. There has to be some carryover effect though.
In my opinion, having a strong core is equally important to having strong legs or a strong upper body. Being able to generate a lot of abdominal pressure/stability when lifting stuff off the floor or holding something/someone in the air is crucial to maintain good posture and avoid lower back pain/injuries.
Now, if you’re currently having lower back issues, I’d lay off exercises that load your spine axially for a while. The goal is to eventually get back to doing these exercises but first you need to fix your lower back without postponing your desire to increase overall strength.
Therefore I’d pick sled work for the lower body because it removes axial loading of the spine to a great degree and puts almost all the work in your legs. You also have to engage your core when pulling/pushing to make sure that your energy transfer comes from your legs and not your lower back. This increases your awareness of bracing, which will come in handy when you move back to exercises that load your spine and lower back.
Sled work also creates very little muscle damage because the eccentric portion of the lift is removed. But it still allows you to grow muscle and build strength, even power. 10 meter equals one rep so after an extensive warmup you could do something like 3-5 worksets for 30-50 meters (3-5 reps) to build strength in your lower body.
When your lower back issue starts to dissipate I would move back to a mixture of sled and squat/deadlift variation work. For example combining a sled pull and a trap bar deadlift where your upper body remains more upright than a romanian or conventional deadlift. Or a sled push with a zercher squat/front squat to maintain that upright position.
For upper body, the focus would be again to increase core strength and upper body strength so you can safely move around when holding things/a person at your side, in the air or on your shoulders.
Loaded carries are an incredible tool for this purpose. Any type of loaded carry will have some carryover but it’s best to pick variations that match daily movement more closely: farmers walk for groceries, sandbag shoulder carry, zerchers carries or overhead carries with a db/bb for carrying your kid on your shoulders etc.
The same distance parameters for sled work apply here when trying to build strength.
You eventually progress to doing a mixture of carries and lifting here as well. For example combining a zercher carry with an incline bench press. Or a standing overhead press with farmers walks. Or a decline pressing variation with an overhead carry etc. Just make sure to select exercises that do not hinder each other too much.
For all lower body and upper body exercises where the goal is to do a set number of repetitions (deadlift, squat and pressing variations) I would start in the 6-8 rep range, using the triple progression model. Which means that you stick to the same weight for each workset and try to get 8 reps in each set. If you can do all sets for 8 reps with the same weight, you increase the weight. You keep doing this until you can no longer reach 8 reps for each workset for 2 weeks straight. Once you can no longer increase the weight, you move up to 4-6 rep range and start the same process all over again. When progression stops, you move up to 3-5 and do it again.
Once you’re stuck in 3-5, go back to 6-8 where you will now automatically be able to use more weight because of the neurological adaptions you gained from training in the 3-6 rep range.
I’d stick to 3-4 worksets for each exercise.
I’d start at 3 times a week for training frequency, for example monday, wednesday and friday. You can move up to 4 times per week if recovery is good. 5 times per week will probably be too much.
Do 3-5 worksets for sled work, 3-4 worksets for lifting work.
Rest periods: 2-3 minutes for sled work, 3 minutes for lifting work.