If you're level 55 and you're primarily interested in the story, then you'll want to go back to the beginning. By which I mean, you're going to want to finish the Belsavis world arc, ideally on both factions. Anyway…
Eternity Vault isn't important for the Dread Masters storyline, but it's still pretty epic. It's by far my favorite op in terms of set pieces and scenery, and the final battle is extremely epic, if simplistic. Frankly, all of the battles are simplistic. You shouldn't have any trouble finding a group of like-minded people to stomp through EV Story Mode with you, and there are very few people who *haven't* done EV at least once. At level 55, you can pretty much 4 man it with no experience. Naked.
Karagga's Palace is the first mention of the Dread Masters (though most people miss it in the cutscene if they aren't paying attention). I won't spoil anything else, just assume that the rest of the ops are required reading for the storyline. Like EV, you can do this one mostly in your sleep (especially in story mode), though from this point forward, you're going to start needing an actual ops group with a real composition (tank/tank + healer/healer + dps/dps/dps/dps). Still no gear requirement.
Explosive Conflict is the first operation with real mechanics. You can actually wipe on this one. If you have a real ops group though with the given composition, you should be able to clear it without too many problems. Read up on some of the guides. No gear requirement, though being in full 66s or better will make everything a lot easier (except for the HP-driven mechanics on T&Z and F&S).
The next op in series is Terror From Beyond. This is the first of the level 55 ops in your journey, and also where things start getting really well designed in terms of mechanics. I wouldn't say it's harder than EC (in fact, in many ways it is easier), but you no longer have a level advantage. I would recommend full 66s before you go in here on story mode. It's very doable in far less, but gear pads your margins and makes some things a lot easier. You'll definitely want to read some guides, or alternatively just have someone in your group who has experience.
The most recent op is Scum and Villainy. The general theme of this op is: more bosses, less trash (especially compared to EC), longer encounters and more creative use of the game engine. I actually like the mechanics in TfB a lot better, but S&V is still fun. As with all of the ops since EC, mechanics are important even in story mode, and it is possible on most bosses for just one person to wipe the raid if they do exactly the wrong thing. Since it's your first time through, it will probably take more time than TfB, but not much. Same gear requirement as TfB.
Run through those ops in the order I listed and you will be completely caught up on the storyline. There are some really good story moments along the way. Frankly, I'm a bit jealous that you're seeing it all for the first time. Soa is awesome, Jarg and Sorno are just fun, Kephess is still one of my favorite fights (and the Republic cutscene after Denova is probably the best in the game), TfB feels so creepy and alien, while Scum and Villainy is different and creative and has a nice twist at the end. There's some good content here, and those of us who see each of these multiple times a week tend to forget how fun it really is. Enjoy it, and don't let anyone rush the experience!
My best advice for finding people to run with you is to talk to some friends. Whisper some people that you've met doing flashpoints/RP/other and see what they think of the idea. If you're on The Ebon Hawk, you can send a mail to any one of my toons and I'll try to set you up with a good group.
Do not use Group Finder for this! The people in Group Finder are doing it for the commendations, and they will more than likely just kick you from the group if they see you aren't in full 72s. Also, running through Group Finder can result in some missed cutscenes unless you know what you're doing.