Ok, so I've never played the AA games. Been semi-interested for awhile. This sale has me close to checking them out. A little background would be helpful.
1. Is each game a continuation/direct sequel? "Trilogy" gives me the impression that it is.
2. If not, would you recommend DD or Trilogy? Even so, would DD be hard to follow without having played the trilogy?
3. Is DLC a must to thoroughly enjoy DD?
4. Ultimately, which is considered the best?
Yes, it has continuity. The trilogy is the most cohesive part, hence why it's called a trilogy. Game 4, Apollo Justice: Ace Attorney has a big time skip and a new protagonist but is definitely continuous.
Dual Destinies is game 5 and has the return of Phoenix as main protagonist. It is definitely after Apollo Justice though and includes him playable as well.
But it also goes out of its way to sort of not really touch much on the events of Apollo's game? The series had a really long gap in mainline releases so Dual Destinies is sort of meant to be a jumping in point, and you would be able to get some enjoyment out of it.
Ultimately though I'd recommend starting with the trilogy and playing the whole thing in order.
The first and third games are probably overall considered the best in the series. The core original trilogy is probably going to remain the definitive bit of the franchise. So I'd definitely tell you to start there.
(It looks like the next game, #6, Spirit of Justice, will be returning to some of the loose threads from #4, Apollo Justice, as well, so it's definitely not like that game is being disavowed. It's a shame there's no easy way to get it right now; that it's not available on the eShop like the other games in the series.)
EDIT: The DLC case is one of the best in Dual Destinies. If you pick it up, get the DLC. It doesn't detract from the rest of the game or anything it's just an extra standalone case.
Ace Attorney kicks ass. Do yourself a favor and get Trilogy and Dual Destinies and its DLC right now, and go track down a DS copy of Apollo Justice since that's the only way to play it. You'll thank me later.