O.K. I will assume that what you have is all you have access to. Try to progress beyond what you have listed. So, if you're squatting 60lbs for 3X8, work up to 3X10, 3X12 and then add a 4th set and so on.
I would not exclude upper body work and other exercises. If you have the barbell, you can press over head. As mentioned above, you can learn to power clean and do pull ups and push ups. You can also deadlift.
For core, planks and pillars are useful. There are tons of different exercises.
If what you have outlined in your original post is challenging and hard for you, then you should make progress. But if the lifts are not that difficult and really don't push you or challenge you, then it's just exercise for the sake of exercise.