Some disorganized notes:
C throw is one of your primary anti-airs. Doesn't work against crossups, but it's very strong against jump attacks coming from the front.
Thunderclap (back A) is slow, but has a good anti-air hitbox. It eats projectiles, including Valerie's Rainbow Disc(!). (But not Geiger's Cycloid Revolution.)
Air super is fast and invulnerable and beats most things air-to-air; it's yet another good anti-air.
Always punish with normal throw instead of C throw if possible; this gives you much better oki. (Normal throw is also much faster.)
Your basic oki mixup can be C throw vs. The Boot. Throw if you think they'll block; use The Boot if you think they'll jump. Either half will do two damage. If you're doing this off of a long knockdown, you can also mix in attempted cross-ups with jumping A. Also, if you don't want to commit to The Boot, you can use neutral A sweep instead, which is of course much safer. (Also, there isn't any timing fuckery that will get you Yomi Countered.)
The Boot is not safe on block if done point blank, but can be safe at some ranges. However, it will still be your opponent's turn to push buttons. Depending on timing, The Boot is a great way to call out people who try to jump away to avoid throws. (If your opponent is throwable and you're point blank, you can cover both jumps and blocking, since if they blocked, you'll throw them.)
Ground B is a little like a jump when you're using it vs. fireballs; you have to predict if you want it to hit, since it's so slow.
Air A has a great hitbox. Better than the old air A I think, and the old air A was quite good.
Air neutral A is great for stuffing ground approaches, and can beat various air things too.
Air B (lariat) obviously goes through fireballs; this is a really strong property. It also has very good priority air-to-air, and it changes your momentum as well.
Air C can be used to avoid anti-airs like Grave's knifehand, by stopping your forward momentum.
B/C/Super's armor is broken by supers and also by DeGrey's Pilebunker.
If you're getting crossed up, try jump back air A, or air super.
That's all I can think of right now that isn't particularly matchup-specific; maybe it can be organized into more usable info and put into a proper guide.