Squats are the best leg exercise. Try front squats and overhead squats as well. The force you to use correct form and help correct back issues.
Deadlifts and its variations are important for proper hamstring development.
Lunges and split squats are great for quad development and also for balance and coordination.
If you start light and focus on technique you can work legs 3 times a week without DOMS. Keeping that up as the weight gets heavier is easy if you rotate the exercises around in a heavy, light, medium pattern.
Back Squats (heavy)
Romanian Deadlifts (Medium)
Overhead Squats or lunges (light)
Front Squats (Medium)
Glute Ham Raise (light)
You can do something similar for upper body.