In order to get started with the raid content in Legion, you need to gear up to somewhere between ilvl 825 and 840, which is perfectly doable without too much farming. It does not even require all that much luck if you follow the guide below, and you should be ready to raid in no time!
The first pieces of gear you want to target is the class gear. These requires different tasks to unlock and get, but they can all be upgraded to ilvl 840 by using the armor kits sold at your call hall, so these are very nice to get. Below are the sources for all the peices:
Belt: Unlocked after earning a total of 100,000 Artifact Power.
Chest: Complete the entire Class Order Campaign.
Shoulders: Exalted reputation with The Nightfallen. Here's a rep guide for the Nightfallen.
Wrist: Unlocked once you have recruited 6 Champions for your Class Hall.
Helm: Unlocked once you reach level 110.
Feet: Need revered reputation with 2 of the Legion factions.
Leggings: Unlocked once you have completed all 8 dungeons that were added in Legion.
Gloves: Honored reputation with The Nightfallen.
Once these are all obtained and upgraded to ilvl 840, you should be somewhat close to being ready to raid, but if you still need more or better gear, you can read the other ways to obtain pre-raid gear in Legion below:
1) Do World Bosses. These show up just like World Quest, and they give you a chance for loot on the first kill every week. Each World Boss has one specific slot they drop items for, such as Calamir which only drop boots.
2) Grind out some reputation. Most of the Legion factions offers an ilvl850 item once you hit exalted, which are pretty good to get before you start raiding. Getting rep can be pretty annoying, but the best way is to combine them with World Quest, which also can award gear which are scaled to your level, so make sure you do them.
3) Buy or craft gear. Crafted gear is pretty expensive, but it's a nice way to get higher ilvl if you have extra gold. If you want to craft it yourself you need a lot of Obliterum, if else you can just check out the AH for the gear you need.
4) Do dungeons. These drop ilvl 830 in the heroic version, so they are not all that bad, but the mythic dungeons drop better gear which is perfectly nice to use for your first raid.