It’s somewhat of a kinda lengthy process you have to do once. You need to rank up your grand company rank to maximum to unlock Grand Company Squadrons.
The fastest way to rank up in the grand company is grinding grand company hunting log, Leveling Roulette with a preferred class, FATEs and guildleves (found in many ARR zones), until you reach the 6th rank (Sergeant Third Class). Once you reached that rank you can get grand company seals via green/pink/blue items that you turn in at the Grand Company. Primal Weapons from Heavensward primals are currently dirt cheap (especially the Sophia weapons. just search for goddess on the MB) and they’re an easy way to cap your GC seals to rank up faster.
Once you have reached the max possible rank a quest will pop up in your grand company to unlock squadrons. Once you’ve unlocked them you need to recruit a new member. You get member recruitments by doing your challenge log (it’s rng). Fastest way would be to go to the gold saucer, win 10 triple triad matches, do a few minigames, and by then you usually have a new recruit.
Afterwards you need to level up your squadron members by sending them away on missions or doing training sessions (can do 3 per day).
Once you can form a normal party (1 tank, 1 healer, 2 dps) with squadron members that are level 20 or higher, you can do dungeons with them, and use some of the Scenarios found in here.