While it's generally the case that you should play what you want to, there may be certain considerations for classes. The good new is that, for PvE, the classes are generally fairly even. There used to be quite the spec consideration required for level 70 twinking, as it was possible to hit the end talent for some specs, where that was a good talent to go for, but that isn't really a valid element any longer.
As far as PvP twinking goes, there are certainly some that are stronger than others, but you should still look to play what you like. If you're looking for current strong classes and specs, this forum thread is a decent resource. For level 19, there are definitely several overpowered specs, check out what's popular, hunters are one specific class that stands out! The game is very hard indeed to balance at level 19.
Gearing your twink
With twinking, gearing is where the fun really starts. For PvE, there are various considerations for the would-be twink. Twink raiding guilds will often impose certain item level requirements, based around twink raiding achievements, as some cannot be done if your raiders are over certain item levels or wearing certain gear. Guilds with these restrictions will generally be pretty clear on them during the recruitment process, as they are so key to their goals.
For PvP, the best gear is the best gear! There is an interesting choice to be made between going for resilience versus damage, with resilience gear being fairly widely available at the moment, but a distinct lack of PvP Power to cut through it. Because of this, the elevated stats on some PvE gear can be preferable to PvP gear to maximize damage output. Level 70 gear is purchaseable very cheaply for honor, while level 80 gear shows a huge increase in cost and is unfavorable at best compared to Cataclysm greens. However, there is a point where the resilience gains from PvP gear outdo the damage gains from the greens. There are several guides available to getting the best gear, such as this forum thread.
Finding fellow twinks
For a would-be PvE twink, this is certainly one of the hardest tasks. There are a few well-known twinking guilds out there, one being iToons, on Korgath-US, being the guild run by the Convert to Raid podcast, and viewable on the forums section of their website. They are currently focused on level 80 content, having finished off Burning Crusade recently, and are running their first Wrath raid on the 6th of January. Twinkinfo is also a great resource, where many guilds go to recruit for twinking activities. There are a few guilds listed on twinking.net, too, and the Blizzard realm forums are often good places to start if you're not interested in realm changing.
For PvP twinking, your life is a lot easier, simply because you only need one friend to do arenas with! If you don't know anyone who's into PvP twinking, there is no real hardship in rolling a few alts on a few servers and getting them to level 19, and keeping an eye out for other PvPers.
Mists of Pandaria is here! The level cap has been raised to 90, many players have returned to Azeroth, and pet battles are taking the world by storm. Keep an eye out for all of the latest news, and check out our comprehensive guide to Mists of Pandaria for everything you'll ever need to know.