World quests should be scaling up with your ilvl, so you should start seeing 365/370 emmisaries soon.
When darkshore is up spam the ever loving crap out of it, that gives 360 gear.
Do your weekly warfront for a 370/400 item.
Do lfr every week for 370s.
Do the two warfront world bosses when their up for 370/400 items.
Once you hit about 375, your emmisaries should start giving 380/385 gear which I believe is the cap right now.
There are lots of options =D, I did most of these and never stepped foot in pvp/raids (technically I did lfr, but it’s when I over geared it so I was there purely for transmogs…)
WQ gear goes higher than that. All I do is log in, do WQ’s, World Bosses, Battlefronts & occasionally Incursions and I got to 389 without even trying. I do LFR once for each wing just to see it, rarely get any upgrades out of that though.
Mythic 0 are almost inexistant on the dungeon finder. Your best bet is to find a guild that runs those and then you can jump in mythic+ or lfr/normal BoD.
Press I to toggle your Dungeons and Raids window. Select Premade Groups > Dungeons > Find A Group. Search for “M0” or “World Tour”, the latter being when a party tries to do all the dungeons on Mythic 0 to load up on 370 gear. You can also start your own World Tour group if you feel up for it. If you queue as a healer you’ll get invites pretty quickly… plus you can learn the dungeon mechanics and then start doing Mythic +, which drops anywhere from 370 to 400 gear depending on the level.
looks like you really need REP with Champions of Azeroth. I’d focus on dailies/emissary’s + a contract for them if you don’t have at least one toon exalted with them.