This is a spoiler-free summary of the systems changes that come with the first major patch of Shadowlands, 9.1. The patch itself is June 29, the raid opens the following week. So, there will be new areas, quests, stories, etc. on June 29, but the changes in gear will happen with the release of the raid.
tl;dr Domination Sockets are good when you can get them. New Legendary levels require more Torghast and will take a few weeks before they matter.
1. Gear changes. The most complicated new system is that the new raid drops special pieces that have "Shards of Domination" sockets. Think of these as a special kind of gem (which also drops in the raid), they each have special bonuses, and if you wear a set of them you get a special set bonus (which only applies in the raid, Torghast, and the Maw). These gems are powerful; estimates are that a Domination socket is worth 50 ilvls on the piece. This may be a way for Blizzard to get mythic raiders interested in running lfr. ) These sockets only hold Shards of Domination, not regular gems.
a. There are 5 potential slots for Domination sockets. Everyone can get them in Helm, Shoulders or Chest, and the other two slots that vary by armor type:
cloth: bracers & belt
leather: gloves & boots
mail: belt & boots
plate: bracers & gloves.
While the main source of socketed gear is the raid, there is a quartermaster in the new area that will sell one piece of socketed gear for your armor type (i.e., not the helm, shoulders, chest but one of the pieces listed above by armor type). In the raid, as usual gear from the last 2 bosses is higher ilvl than the rest, so socketed gear from them is 'better.'
b. There are 3 families of gems: Blood, Unholy, and Frost (because Death Knights really do rule the Shadowlands). There are 3 different gems in each family, each with a different effect. The gems are unique-equipped, you can only have one Blood Shard of Bek equipped for example. If you socket 3 gems from one family, you get a "set bonus" (Runic word)--although to get it to activate, you have to have one of the gems in particular armor pieces. E.G., the Blood set bonus will only activate if a Blood shard is in a helm, Frost requires the shoulders, etc. Oh, and just for additional fun, each gem, er, shard, comes in 5 different levels from "normal" to "Portentous". And you can increase the level of the gem with a different currency that drops in the raid...
c. If all this seems complicated, it feels like it's complexity to make it look busy. You want to equip a piece with a Domination socket any time you can get one. At the rate we go, you're not likely to be deluged in gear, so it will be a while before you have even 3 sockets. ) It's not clear what the drop rate on the gems will be, you'll either have just one or two gems to choose from or the complete set and can pick the best of what sounds good. There is an item you can get for fairly cheap that lets you take the gems out and move them to different sockets, so you can change your mind. When you get enough of the new currency you'll want to upgrade your favorite gem. Tanks probably want the Blood set bonus, DPS probably the Frost, and Healers probably the Unholy, though specific classes may have different favorites. (I don't know, Healers may not care about any of the set bonuses.)
d. Do note--mythic+ gear does not have sockets. So, if you are looking at your Great Vault rewards, you probably don't want to choose a piece that goes in one of your 5 potential Domination Socket slots.
2. Legendaries--You can now "upgrade" your legendary to ranks 5 (ilvl 249) and 6 (ilvl 262). There are some small quests to do for the Runecarver to activate this. The new levels require a new currency, Soul Cinders, that you get from the new layers (difficulties) in Torghast (layers 9-12). As usual, to upgrade you need a new base item of the right level (crafted or from the Auction House), the right number of Soul Cinders, and possibly Soul Ash (if you only had a Rank 2 legendary, you'll need the soul ash to go from rank 2 to rank 4 in order to skip over them). Note that if your current legendary is in one of the 5 armor slots that can have Domination sockets, you probably want to move it to a different armor slot (Blizzard added new choices to things that could only be crafted in one of the 5 choices). In that case you'll also need to supply two new missives, you are basically crafting a new legendary. When you do a run that awards Soul Cinders, you also get all the Soul Ash as well.
a. In addition to Torghast, there is some other content available that awards Soul Cinders. If you do absolutely everything possible, the soonest you could craft a Rank 5 legendary is about 2.5 weeks, with about 3.5 weeks for a Rank 6. In the time it takes to get the required Cinders you'll have earned enough Ash to build a new legendary. (For me, that's more Torghast than I can stand, it will take me longer than that to get there.) For reference, Torghast level 9 (the first one that awards Cinders) is recommended for ilvl 208.
b. We have some guild crafters who can build rank 3 and 4 legendary blanks. There's a recipe the crafter has to buy in the new area (requiring reputation, of course) that lets them make an item that increases the level of a legendary they make by two tiers, so they can make a rank 5 with the rank 3 pattern. Rank 3 only requires a horrendous amount of materials, not stupefying. The new rare material needed to do these upgrades is Korthite Crystal, a drop from rare mobs/treausure chests/daily quests/etc. in the new area. They are tradeable, so there it likely to be a lively market in these at the start of the patch.
c. There are new memories (legendaries) available for your Covenant.
d. There's now a way to transfer soul ash to and from alts.
3. There are lots of character abilites changes. Tanks still tanks, healers still heal, and dps still, well...
4. Starting with the release of the raid, ilvl rewards from mythics and mythic+'s will increase (mobs will also hit harder). There is also some ilvl 200 gear available in the new area for alts.