I know what you mean about feeling beat up from starting strength, it happens when you're beginning and especially due to squatting 3x a week. The program itself is not bad at all, just make sure you practice proper form, eat enough, sleep well, etc. do not underestimate these. Also, the best tip I can give you having been there and done that with the whole 'beginner phase' of starting strength and linear progression based programs is this:
Never train to failure on any exercise or any set.
ie. for a 5 rep set, always use a weight that you can do 7-8 clean reps with or
similarly, for an 8 rep set use a weight you can do 10 clean reps with, etc.
This will dramatically help your recovery immediately and won't leave you feeling beat up, instead you'll be ready to go at it again sooner than before. This is especially important when you are frequently squatting/deadlifting/pressing/rowing as those movements definitely take a lot out of you, even more so when you are constantly trying to add more weight to the bar. You will also find that you will not hit any plateaus as soon as you did before while doing this. Additionally, remember not to get too caught up in the whole linear progression scheme, if you are not feeling it in the gym which will happen or if the weights feel heavier than usual, back off. Only move up in weight when you can do the required reps cleanly, DO NOT perform any "grinding death reps" where you are struggling with the weight throughout the set and compromising form. You will only dig yourself into a very DEEP hole by doing that.