Is this a good training routine?

Age, height, and weight?

If you are of high school age, squatting 60 lbs for 3 sets of 8 is not going to do much. If you're a smaller sized middle schooler who will be in high school next year, and this is your first venture into lifting weights, then it looks good.

14, incoming freshman, and I'm almost 5'10 and about 123 lbs. 

Too much plyo (only 3 days/week, not every day).  I also don't see any core work. I also don't see hang cleans which is one of the three core lifts (squat, bench, hang clean). For the strength work start with 2-3 sets of 10-12 reps.  You want to feel the the burn on the last two reps of each set - no burn, add weight - too much burn, decrease weight.  As you get stronger then you can look at lifting a higher percentage of max 1RM (so 3 -5 sets of 5 or a 5,4,3,2 progression, etc).  Your first goal should be a basic level of stregth which would be a bench of 123lbs and a squat of 246lbs (hang clean somewhere in between).  A couple of great strength exercises that take no equipment are pull ups and push ups.  There are many ways you can make these two harder if you want   You also need to incorporate lunges, step ups, and medicine ball work into your strength workout as these are fundemental movements for baseball.  As for speed, include form and technique work (starts, skips, bounds, etc) and not just sprints.

If you haven't lifted have someone teach you.  Many YMCA's have a program for young adults where they check you out for weight room use and also teach you how to lift.  If the YMCA isn't for you then find a strenght program you can join.  Volleyball clubs, soccer clubs, basketball clubs usually have these as well as many communities.

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.