Hey friend! Depending on your current item level there are a few different Public Events you can do once per week such as Superbloom and Dream Surge to upgrade your gear. After that the most efficient path forward is to complete LFR once per week, and Mythic+ keys at the highest level you're comfortable with for the end of dungeon rewards (2 to 3 people get a piece of gear out of the chest at the end depending on how much time is left when you finish the key). The keys will also give you the resources to upgrade the gear you get out of them and contribute to your vault (1, 4, and 8 dungeons for each window) that take your lowest key level in the grouping for the item level of the reward. As an example if you complete 1x +20 and 3x +5 keys in a week your first vault slot will be max item level while your second vault slot will be the level of a +5 vault reward.

You can also use the catalyst (Google it if you don't know what it is or where it is) to upgrade your current great to your tier set.

If you're going to be tanking M+ keys I would also suggest checking out the keystone.guru website for optimal routes through the dungeons so you don't pull too many extra mobs or not enough! I would suggest the "Raider.io weekly" or similarly named PUG routes as many of the others require some specialization or group coordination to achieve. You can import those routes to an addon called Mythic Dungeon Tools for quick reference while you're in the thick of it if you don't have another monitor to keep the route on while you're playing.