7.1 has three major features. There are a few others that bear talking about, but let's talk about the headliners, and what you should do to participate in them.
Return to Karazhan:
You cannot run this dungeon until you have attuned to it. I believe that you only have to do so on one character, after which any alts will be free to run it too.
The quest chain here starts in Dalaran. Khadgar will send you to kill God-King Skovald in Mythic Halls of Valor (you'll probably want to beat Odyn as well, but I don't know if that's required for the quest.)
Subsequent quests will send you to do Mythic Black Rook Hold, Mythic Assault of Violet Hold, and Mythic Vault of the Wardens. After this, you should be able to run Return to Karazhan (which has a recommended minimum item level of 840.)
Trial of Valor:
Wait two weeks! The Trial of Valor raid is not yet open. However, we will (apparently) be getting world quests all over the Broken Isles where we fight Helya's Helarjar forces.
The Suramar story continues. If you have not yet finished the Suramar quest line, you'll still have to do that, as these quests follow directly from that main storyline. Given that if you've finished those quests you'll probably be either exalted or very close to exalted, the gating on these new quests is simply time - you'll get new ones each week.