If you're new to Path of Exile, welcome to Wraeclast, and I'm sorry nothing makes any goddamn sense. While you're working out the difference between "more damage" and "increased damage," here's a quick writeup on how to get the most out of the game's cash shop.
Is Path of Exile pay-to-win?
You can buy stash tabs as permanent account upgrades. Some of them will auto-sort certain classes of items, or make player trading a little easier. In the endgame, where monsters explode into massive piles of miscellany, they can save you some headache; less time spent managing your stash is more time spent farming, so you're going to want a small set of them if you get serious about the game.
Everything else is cosmetic—and it's really cosmetic, no costume stats or hidden sockets.
The answer, to make a long story short, is no. There is a marginal buy-in if you want to guarantee that your stash-management delays are minimal, but top-end looting strategies and ladder play are possible whether you spend money or not.
What do I buy first?
Nothing. Finish Act 10 before you even consider spending real money.
So what's the deal with Supporter Packs?
Supporter Packs are bundles of themed goodies that usually come with points to spend on whatever you want. You can find them in the in-game store by clicking "Buy Points" and scrolling down.
If you're going to spend $20 or more, you want the First Blood pack, no questions asked—it's a "free" stash tab and weapon effect. The other packs are more focused on their cosmetics, and while the lower tiers are usually good value, it depends on the contents.
Wait, these prices are completely fucked up.
Yeah, most cosmetics (and the higher-end Supporter Packs) are priced for whales. C'est la vie. If you're gonna buy cosmetics, wait for them to go on sale, or pick them up as part of a Supporter Pack. You can also get some pretty nice-looking stuff for completing league challenges; hit H in-game to take a look at whatever's being offered.
Stash tabs, on the other hand, are pretty fairly priced considering how much time they save. It's unfortunate that they're not free, but they're a one-time purchase and I definitely feel like I've gotten my money's worth.
Okay, I'm in the endgame. Now what?
The biggest quality-of-life upgrade in the game is a Currency Stash Tab ($7.50), which also happens to be the highest-capacity stash tab in the game. Without one, storing and organizing low-value currencies becomes a huge pain in the ass—to the point where you might not even want to pick them up.
If you're not in SSF, at least one Premium Stash Tab should be next in line; they let you easily list items on trade sites, instead of using a utility like Acquisition. The Premium Stash Tab Bundle (6 for $20) will expand your stash to 10 total tabs, which is pretty much the perfect amount for endgame comfort. If you're on a budget, you can stick to standard tabs, and upgrade a single tab ($1.50) just to have access to low-friction sales.
After that, you can pick up the other specialized stash tabs in whatever order you find convenient—I highly suggest the Map Stash Tab ($15) next. The Divination Stash Tab ($5) is your lowest priority, since most divination cards are low-value trashgarbage intended to make items accessible in SSF. You could literally remove divination cards from your loot filter (you are using a loot filter, right?) and almost nothing would change; some of them are valuable, but the valuable ones are very rare.
The Premium Quad Stash Tab ($15) is useful for long-term archival, the chaos recipe, or a massive "it'll sell eventually" stash, but if you're trying to manage multiple types of items, it can get clumsy. Don't make it a priority unless you have a specific use for it. Stash tab bundles are more efficient, and easier to manage, if you just need more space.
Standard Efficient Spending Strategies apply. Don't buy garbage just because you have some points left over; new stuff is released frequently and you might want some of it.
My character is wearing a pan for a hat. Please help.
If you really want cosmetics, my recommendation is to focus on stuff that any character can use. Portals, footprints, and armour are the big ones, but a lot of skill effects will be useful on multiple types of characters, like Heralds and movement skills. Weapons are pricey, but some of them look alright, and weapon effects can be a nice way to give any character some flair on a relative budget.
Avoid pets and hideout decorations unless you really want one and have unhealthy amounts of discretionary income. Pets are practically invisible in the hailstorm of particle effects that many builds generate, and most hideout stuff looks like shit—sometimes it can even cause speed-bumps when trading, because your 60 stacked particle effects are torture-testing the other player's PC. Character Effects are a tossup; some of them look like shit, some of them look okay, and they're all a billion dollars so why the hell would you even bother.
Mystery Boxes are deceptive bullshit. The individual items will be on sale in the shop later. The price of a box is the same as the cheapest item it might contain, which might seem nice, but you can and will get duplicates; cosmetics don't bind to characters, and they're trivial to retrieve, so if you get a duplicate it's basically throwing your money away. Avoid these unless you're 100% certain that you want every item in the set, and even then, only buy one.
The most important rule: if you buy wings, you're an asshole.
Happy farming. Stay sane.