Some people consider 22 LR to be the ultimate ammo for preppers. It's widely available, it's still relatively cheap (even with skyrocketing ammo prices), and you can easily carry 500-1,000 rounds of it in a bug out bag. A 22 LR rifle is great for varmints and small game, and in a pinch could be used for personal defense with a well placed shot if you know what I mean.
It's likely that 9mm is the most commonly used caliber of all these days. With so many great handgun options on the market ranging from subcompact all the way up to double stack 1911s, it's obvious why it's become the king of the EDC. Finding 9mm after SHTF should be a lot easier than most other rounds given it's such commonly used ammo and so many people carry these days.
Perhaps the greatest creation of all time, 5.56 is a fantastic round and definitely something you want to have stockpiled. It's an incredible round for self defense and in a pinch you could hunt with it as well as defend yourself from any predators. The options for rifle platforms chambered in 5.56 are absolutely limitless. The value of this round in a short or long term collapse can not be overstated.
A great round for hunting, .308 will likely tackle any task you have at hand. While it's more commonly used by folks in bolt action rifles, you do have other platforms available such as the AR10 and LR 308. The ammo is decently priced with average availability and tons of options as far as grain and quality. You'll definitely want to have this on hand in a post apocalyptic landscape.
A shotgun is not my first choice when it comes to weapons, but they do have their place. They are devastating when used for home defense, and certainly something you want to have on your shelf. Shells are super cheap and you can find them damn near anywhere.
300 Blackout is quite possibly my favorite of all rounds. It's fun to shoot and it is great to run suppressed. It didn't make the list only because it is more expensive than your average ammo and sometimes it can be hard to get your hands on.
People refer to 45 ACP as God's caliber, but most guns that use it have a low capacity and therefore it doesn't lend itself well for survival.. however in a SHTF situation it may be widely available through scavenging.
7.62x39 should be obvious. Most AKs are chambered in it and they absolutely fuck. That's about all that needs to be said about that.
350 Legend makes the list just because it's a nice game cartridge and in states that have restrictions on ammo, it may be one of your only options.
I love 6.5 Creedmoor. It's an extremely accurate round that shoots well and at long distances. The downside is that it can be difficult to procure and the price usually comes at a premium. However, like with 300 blackout, if you can find it and afford it, do it.