You should be able to anti-air the moonsaults. I don't know what Gief's best button is for that, but Lariat should work, I guess.
Standing jab should smack the moonsaults, right? I don't play Zangief, but that button is pretty great.
Easiest punish for moonsaults is Jump+you best Air normal as an AA.
Standing Jab or any other traditional AA is risky because Moonsault has a very weird speed at which it attacks. Nash jumps up quite slowly but he comes down really fast. So if you whiff your AA or time it too late , you are going to eat a combo.
Jumping+normal is your best option since at worst case you trade and air reset.
Also since you are gief I would also experiment with the Armored punch move , It might be a bit slow but its worth a shot.
If Nash ever dares to moonsault air SPD or lariat it.
Aside from that...its pretty much as you said.
Your best bet will invariably be to get vtrigger and try to use it to get damage off a punish or forcing a situation with suction. Other then that hope for luck with parrying and pushing Nash to the corner and pressuring as well as you can.
Against a really strong Nash though, expect to eat a lot of booms, "get lost" kicks or backfist. Patience is vital here, but you really need to grasp the few opportunities (jumping over predictable boom patterns or scythes, punishing bad anti airs) to win.
Expect to lose like 75% of the time tbh
Nash is afraid of Gief. Every character should be, but Nash especially. First of all, I believe your f+HP headbutt stops projectiles. You also have v-skill which can be used similarly (along with the standard forward and neutral jumps). Switch up how you react to the booms to get them to dash/jump in. Moonsault can be stopped with a standing jab (go to training mode and set the CPU to do moonsaults over and over until you can jab it on reaction, it will make your life a lot easier). Gief can probably stop it with whatever his anti-air elbow move is (cr MP?). Your V-Trigger should eat him alive.
Once he's in close, I think his fastest normal is 4 frames so you can jab or tick throw (jab, pause, command grab) him to death. IMO a well played Gief is a nightmare for every character, but Nash probably has a slightly harder time than others.
Edit: for Laura, learn to identify the difference between her light elbow and her medium elbow. If they do light elbow, you are going to want to risk some neutral jumps after to escape pressure. If they do medium elbow and you block, you can punish it pretty hard (command grab). She has to be close to deal damage, and that's where you want your opponent. Just be comfortable with landing a few hits (cr jab, anti-airs), and know that it adds up quickly, and once the health drops opponents tend to get a little reckless.
You can Air SPD the moonsaults. When I am beaten by a Nash, they've rarely moonsaulted.
Thanks for all the advice! For some reason I didn't think to try and punish moonsaults with AA normals, lariat or air SPD, but I'll definitely give those a try next time I go against Nash. I did manage to punish them with standing HP quite a few times, but mistiming it leaves me completely open. At one point I actually managed to v-skill all three hits of an ex moonsault, but failed to punish afterwards...