Lazurite (used to be called Gems - if you read Gems somewhere, think Lazurite) comes from camps, specifically Ore Mines.
It is also in Item Store Chest of Materials.
The only reasonable way to get materials is to hit level 6 assailants. Aside from the specific materials per assailant, noted elsewhere, camp materials seems to be common across all assailants. Particular camp materials seem to be more prevalent with certain assailants, but I have no data on that.
[If you find you're not getting a camp material from a particular assailant, but do on others, focus on the others.]
If you work the assailants long enough, you will inherently acquire what you need. It may take a while. Continue working on other assailant materials in the mean time - camp materials will accumulate as part of that.
There is no point hitting less than 6 - which gives Rare through Legendary. Each level below 6 gives one gem level below - so you need 4 of any level below to get one of that level. For the same reason, do not bother with anything below Chest of Materials 4. (4 x Chest 3 = 4 x 4k gp, 1 Chest 4 = 8k gp.)
Ultimately, it won't make any material difference whether you get a material now, or wait until you can hit L6 assailants. Whether you get a material today, tomorrow, or next week, probably won't make much of a difference in your game. Individual differences and improvements being so marginal. Which is to say, don't get too worried if you don't get that gem today - there will be other opportunities sufficiently soon enough.
Work the assailant studies until you have Assailant VI and all above it, except Tactics II, completed. Don't bother with Stamina Restoration unless forced to, and leave Enhanced Attack for last - they will only come into play once you can hit 6.
Don't worry about gear. At all. Early on, everything around you is so overwhelmingly overpowering, no armor will matter. So save your strength.
Once you can hit Ass. VI, get Legendary Savage gear - it makes hitting Assailants -much- cheaper. Then go for whatever gear and material strikes your fancy. Best is to go for Legendary Amarian Build then Study gear. 99 week builds and studies will go down to something like 1/3 of that.
Assailants: https://plarium.com/forum/en/throne-kingdom-at-war/fan-guides/70089_test-s-assailant-guide/?post=382331
Camp Materials: https://plarium.com/forum/en/throne-kingdom-at-war/fan-guides/68627_yielding-materials-tab/?post=379417
Assailant Materials: https://plarium.com/forum/en/throne-kingdom-at-war/general-questions/39829_throne-assailants-jewels-and-materials--/?post=514728
Throne Help: https://throne.help/