Now that I've had my fun, let's get serious - how would I play a rogue at 90? Or a mage, or a priest, or another class that I'd never played before? Well, there are resources out there -- WoW Insider has a rookie guide, for starters -- and there are other sites like Wowhead and Icy-Veins that can and do give you an overview of what your new class does.
There's also a very useful resource built into the game itself. Your spellbook has a great deal of information for you about what your class can do and how to go about playing it, that can get you pointed in the right direction to begin play. We're assuming you're a returning player who has not played in a while or a new player just getting started with this article - established players starting a new alt may already know much of this.
The Core Abilities Tab
The first thing you should do is open your spellbook anyway -- it's where literally all of your spells and abilities are -- but don't immediately start reading them all. You'll do that, but first, go to the Core Abilities tab and check out what's there - there will be a list of your most commonly used abilities. You can drag and drop these abilities onto your bars from the tab itself if you wish to (useful for setting up new bars) and short explanations of each ability and its purpose are given. For example, for an arms warrior you are told which of your abilities generate rage and which ones cost rage and given brief instructions on when to use them. It's not detailed -- there's no explanation of Slam's interaction with Colossus Smash, for instance -- it's just bare bones information that helps you get started. If you're completely unfamiliar with the class you've boosted, take a look over that tab so you get the basics down. You'll need to do some bar customization. Some of your abilities will be placed, but not all. And as for talents ...
The Talent Window
When you level a new character to 90, the process does not select talents for you. Therefore, you're going to have to do that yourself. Wowhead has a talent calculator you can use to take a look at the various talents and get an idea of what you're interested in being able to do. Once you've selected talents in your talent window, they too can be put directly onto your action bar. If and when you inevitably wish to change your talents, you can do so either at a class trainer or by purchasing a Tome of the Clear Mind (they can also be crafted by Inscription). You will have to place these talents on your bars manually, since you will not have them when you first log in after the boost.
Check your gear
The gear you start out in is of a reasonably high ilevel, and is tailored to your class specialization (so a boosted 90 retribution paladin won't start out in intellect plate) but it's not optimized - it has not been reforged or enchanted. Before you immediately leap into LFR with it, you may want to take some time and make some gold (if you don't have any beyond the 150g you start with) in order to reforge to hit your class' target goals. Again, our Rookie Guide or other sites may help you figure out what your target numbers are (for instance, for melee you usually want 7.5% hit and expertise) for reforging, gemming and enchanting purposes.
Basic things you also do not have include most flight points - you do have epic 280% flying, so you can fly around collecting these points.
Read your spellbook
Reading through the various descriptions on your class abilities and talents in your spellbook is something we tend to take for granted on classes we've played for a while, especially after 89 levels or so. But on your fresh 90, you're at least likely to be somewhat unfamiliar with the class and your spells. It's worth taking some time to read them, especially on abilities that aren't rotational but which can save your life or serve as a cooldown of some kind like Ice Block or Mirror Image.
Also check the What has changed? tab at the bottom of your spellbook if you're a returning player who has boosted a class you were familiar with once - it explains many of the Mists of Pandaria class changes to you.
This isn't all you need, of course - the biggest thing that will help you familiarize yourself with your class is to get some playing in. You'll want to at least do some soloing first, get familiar and comfortable with your rotation and cooldowns, and do some work on gear optimization if you intend to be somewhat conscientious when you start queuing for dungeons, LFR and heroics. If you have some 90 friends who are willing to help you out, all the better. Getting familiar with both solo and group play on a class is most often a matter of doing it, but at least with these ideas in mind you'll have a sense of where to get started.
Good luck on that new (sort of) character.