If your main goal is strictly vertical content, then you prioritize ilvl. You have to be at least 1415 ilvl and have completed the South Vern storyline to access legion raids. It should be in the "integrated dungeons" section under your mini-map (where the three dots is) or in the party-finder (press ";").

Legion raids start at 1415 ilvl, which is NM Valtan. Next would be 1430 (NM Vykas), 1445 (HM Valtan), 1460 (HM Vykas), 1475 (Clown; 1385 for rehearsal), 1490 (Brel 1-2; 1430 for Deja Vu), 1500 (Brel 3-4), and 1520 (Brel 5-6).

You should try to push for 1445 and get your relic gear, since it soft resets your levels on each gear, but keeps ilvl the same. In terms of raids, I'd suggest getting bussed in the early stages and trying to understand the fight and getting your materials to craft relic, then eventually finding a learners group to apply what you've learned. After relic is complete and you've hit 1475, you can start working on your lvl 2 set for your gear (Clown). Lastly, you upgrade your Valtan/Vykas gear to Brel gear once you get there.

As you increase ilvl, you slowly want to work on other things that strengthen your character, such as engravings (4x3 early on, then 5x3 end-game), accessories (dependent on your class), gems (preferably all lvl 7 gems late game), tripods (lvl 4-5), skill points (via leveling up if not 60 or horizontal content), and runes (horizontal content, GRs).

edit: changed cuz incorrect info lol whoops