With some delay, here is the entry on Janissary. aelf, feel free to edit it as I'm sure parts of it are superfluous to the purpose of this thread; once started I just couldn't stop.
And if anything I wrote isn't clear, either post here or drop me a message.
Edit: Ouch, waaaay longer than I expected. Sowwy.
Ottomans: Janissary (Musketman)
One of the most interesting Unique Units, the Janissary comes at a time when you're happily building your Universities and are on the verge of getting those nasty Grenadiers to obliterate your last neighbors... That is, if you get them the "usual " way, through slow teching toward Gunpowder.
Part 1: Beelining for Gunpowder
To make full use of the Janissary's bonuses and lack of early counter a beeline for Gunpowder is in order. Whether you take the "southern" path through Metal Casting -> Machinery & Monotheism -> Monarchy -> Feudalism => Guilds or the "northern" path: Maths -> Currency (Priesthood?) -> Civil Service -> Paper -> Education, you can make it to Gunpowder in time to face mostly Longbowmen. Different lightbulbing techniques are available here (GMs for the southern path, GSs for the northern one) so you won't have to self-research everything under the sun.
Part 2: Strengths and weaknesses
The Janissary's UU bonus is a 25% boost against melee, archery and mounted units - so pretty much everything your opponents will throw at you with the exception of siege units. Counting the fact that Pinch is now only available with Gunpowder the Janissaries are quite safe. You'll also get lots of promotions from their first fights (take out lightly defended cities first, forget going for the "prime estate"); in this case going the Combat I + Cover way for most and Combat II in view of Formation for others renders them even more powerful.
The first real threat are Knights, since by then the opponent is one tech away from Gunpowder and will most probably promote them with Pinch right out of Barracks & Stables. By that time though you should already have Pikemen and be on your way to Grenadiers.
Part 3: Getting more!
Costing 80 hammers (the same as the normal Musketman) they aren't very easy to get. Considering that most of the time you won't have early wars while teching toward Gunpowder you'll have only a bunch of cities to help you build your army. Fortunately you won't need a lot of them at first. Make sure to get Construction too for some Catapults, if only for stripping down the city defenses. Once you capture some big cities though don't hesitate to whip the Barracks and then Janissaries from them. You'll be at war for a long time so the war weariness will be quite high in newly conquered cities anyway and you won't be able to make everyone happy.
Another way to get more is to draft them. If you're going the Education route, you're not far from a Liberalism -> Nationhood grab. You can even consider getting another Great Scientist out to lightbulb Philosophy, whether after Gunpowder (to start on the army first) or before (to run Pacifism) and then you're all set up for Liberalism. Of course you can also delay Liberalism to get Chemistry and Grenadiers with it.
A post-patch (2.08) change now makes the Education route even more desirable: the newly improved Military Academy requires Education and gives you a 50% bonus at building military units. Coupling that with Heroic Epic in a high production city will help you get a constant string of Janissaries to keep the attack going.
Part 4: What about the "infrastructure"?
You might be tempted to stop after one or two wars to build universities, grocers, banks, courthouses, etc. Well, do that and you'll lose all your military advantage. Instead, consider changing targets if you're heading too deep into enemy territory. Also, accepting capitulated vassals can really help you dominate your continent at a very early date by 1) not having to fight to the end, 2) getting some forgotten techs from the capitulation, 3) enlisting someone else in your fights. Once you have your own continent you can do whatever you want with it; plenty of time for infrastructure.