I'd like to know how they're normally played
The biggest misconception about Polynesia is that because their Uniques are so...unique that that would limit their playstyle. That's only true if you make it true. I mean, just look how different the above responses are (and wait for how different mine is). Now, having said that obviously there are certain styles better than others and in the end it comes down to the style you are most comfortable with.
what victory types I should go for
I would say any victory type other than Domination is possible as long as you play defensively. (I mean, domination is always possible with every civ as Gamewizard makes a fine case above, but I just don't favor them for Domination games myself. On archipelago maps however anything goes, even domination). Why defensively? I'll get into that in a second. Diplomatic is probably my favorite victory type with them, especially when Greece is absent from the game. At the end of the day it really doesn't matter too much
It's best to settle only coastal and to have a line of coastal cities close enough to each other. On archipelago that's easy. On continents it's about a 50/50 draw of luck. A lot of the time you will have the resource/food availability that favors an all-coastal empire but a lot of the time you won't. I figured archipelago is more straight-forward so I'll mostly be discussing Polynesia with continents and continent-like maps.
The reason coastal is better is for the moai statues which means defense. +10% bonus combat strength near moais may not be amazing, it may not even be noticeable at first, but it absolutely gives you a boost. Princeofnigeria mentioned "Spirit of Adwa" and he's right. To elaborate, Ethiopia's UA Spirit of Adwa gives you a 20% bonus for civs with more cities than you. Polynesia, in some ways, is exactly the same only you get the bonus while being able to expand quickly. But before you tell me 10% is a lot different than 20% keep in mind the secret ingredient: Maori Warriors.
Indeed, Gamewizard mentioned building at least 4-5 - and he's right. Maori warriors give the opponent -10% strength so if you position your troops right then you are basically talking about the Spirit of Adwa with a coastal twist. Absolutely solid for defense. The strategy of that depends heavily on the situation but basically you want to have your warriors on the front line surrounding the city being attacked. The reason you need more maori warriors is so you can rotate them - have a few taking the punches close enough to the moais to receive the combat bonuses, have one or two healing up in the back. Archers are of course great to have but I've had games where the maori warriors and my city bombardment were all I needed - and in those games they cleaned up quick.
It's best to build a maori warrior straight away. That way you have two of them exploring the map, meeting unreachable city-states before other civs and thus getting more cash and if you're lucky more faith. You nab up ruins before other civs (the only civ other than America that gives you this advantage) and often times because you nab up more ruins you find a culture ruin, justifying the decision to build a maori warrior in place of a monument or something else. (Don't waste time/coin building a scout at the beginning...just don't).
Religion. If you manage to get one then take a look at your water tiles. See enough fish/whale/pearl/crab resources? Then choose God of the Sea, provided there isn't a more obvious choice available (such as Desert Folklore if you're surrounded by desert.) Production you get from that is great. By the time you reach oil if you are lucky enough to have an oil resource in the water you are talking about a six tile production (with harbors and seaports.)
Social Policy really depends on how you play but Commerce really justifies settling on the coast for these policies: Merchant Navy (+3 production, head straight for this if you don't have any high prod cities) and Trade Unions (Harbors and Seaports gain 1 gold - this is important too). All of this combined with the the moai gold that hits you at Flight and you are talking about a ridiculous amount of cash (and you can further abuse this money-raking by choosing Tithe if you really want). The great part is, because of the moai culture you will reach these SP's more quickly than your average civ.
You didn't mention any specifics to your game settings but I made a deliberate point to keep everything vague enough to apply to any kind of settings. At the end of the day I call Polynesia my favorite civ to play but I actually don't think they are a powerful civ - not weak, definitely not weak - but by no means part of the top tier in this game. I just like them for the music, immersion and all that
Edit: Looked real wordy, boldfaced for some easier reading