Your beer is done fermenting, but it may not be done maturing. Yeast will continue to break down higher alcohols and eliminate off-flavors even after the end of fermentation.
Personally, I ferment for ~21 days before doing anything to ales, and after that I usually go straight to keg. If you are conditioning in the keg, you'll need to provide some priming sugar to produce more CO2, then wait 21 days at 70F or so for complete priming.
If you are force-carbonating, you can hook up to gas at temperature and pressure and carbonate in a week. You can even carbonate within a few hours if you really want to.
Here's a link to the wiki entry on kegging, which is a good jumping off place for info on carbonating. Kegging - Home Brewing Wiki
Let me just add this: A conical for your first batch? Wow. Nice present.