One tip, don't overthink the tones. Just don't. It seriously kept me down when I started learning. I was all into the sandhi, making sure I got every tone change down and all. But it just doesn't work that way. You have to listen, listen, listen and internalize the tones. You can't just learn them like you would normally. Because when you're speaking, tones merge differently anyway. Listen to any speaker and you'll notice it's never entirely the same unless the subject is being stressed specifically.
As for the combination 3-0 and 2-0, yes there is a difference. The 3-0 combination is like a normal 3-x but just slightly shorter. The 2-0 is basically just a 2-0, nothing fancy. Again, don't overthink it, just listen to a lot of examples and record yourself saying it.
Yes, you have to learn the tones for words, but don't learn the tones as if they're some sort of grammar rule. Funniest thing is, depending on the accent, tones change too. Even simple things like 你吃什么? can change drastically depending on where you are. I've heard people in Hebei pronounce shen2me0 almost as if it were shen4me2. It's hard to really pinpoint but noone actually speaks the way you hear on beginner's tapes.
I wish someone gave me that tip early on when I was going mental learning all the tone-changing combinations. :P