Progression after MSQ

[Stepjam]

Basically from here til the beginning of 7.0, you are going to be on a gear treadmill, frequently upgrading your gear for higher ilvl gear.

Right now you have a few options. To start, you can build up tomestones of aphorism for the ilvl 570 moon gear. This is purely a stopgap measure though, and this gear should be replaced as soon as possible.
If you have money or crafter friends, if you can get a full set of ilvl 580 Classical gear (HQ, never get normal quality gear outside glamour use ever, it's a waste of money), that'll keep you set for a good while.
Once you have high enough ilvl to access it, you can run the normal raid series for tokens that can be turned in ilvl 580 gear. This is roughly the same strength as crafted gear and it's "free", but you can only get so much of it per week.

After that, you have tomestone of astronomy gear. This gear is limited per week as you can only get 450 tomestones per week and most of the gear costs more than that. But it's ilvl 590 and eventually can be upgraded to 600 (either through savage or through alliance raids or hunts during 6.1). This is sort of a long term goal, but one you should be working towards every week.
After that, you have gear that drops from savage raids that are all ilvl 600. Savage as noted also drops the upgrade item for the astronomy gear from certain fights.

For now, those are you gearing options. When 6.1 drops, the alliance raid will drop ilvl 590 gear as well. And then when 6.2 drops, there will be a new set of crafted gear, a new normal raid set, and a new tomestone and savage raid set of higher ilvl. And thus the treadmill grind continues.
Be aware that you never need maximum ilvl to access anything (outside ultimate content whenever that drops, but that's the least of your concerns right now). Having a full set of crafted gear should be able to get you into everything until 6.3 I wanna say when the second alliance raid comes out. And you'll have plenty of time to get better gear than crafted before then anyway.



As for actual content to do, right now as noted, the progression of gameplay is basically Normal Raids> Extreme Trials> Savage raids in terms of difficulty hike. Normal raids are basically the same difficulty as trials you've already been doing, maybe a touch harder. Extreme trials are the game's midcore content. They drop decent gear and if you are lucky a mount. They are a good first step into getting into harder content, at first they'll seem overwhelming, but as you do them more you'll see they aren't too bad. And then Savage is where the content starts getting pretty hard. But it also gives the best gear you can get at any given time.
I'd just start with the current normal raids. You shouldn't really practice on old content as you won't have your full rotation. You should check it out because it's just fun to do though, there's some pretty fun content in the past expansions.