So I've been doing a lot of the combat and have reached level 30 with two of my sylestis, and I noticed that we can obtain 'rank ups' for certain attacks. However, I didn't see anything talking about how we get these 'rank-ups' in the tutorial and I was wondering if anyone could tell me how.
Thank you!

it will level up by itself when you use it a certain amount of time ^-^

Pet attacks only rank up in the Lost Grove- the post-lv60, post-current-main-plot content- and nowhere else.

Once you reach the LG, battling in it will give an attack small amounts of exp for use / the pet being in a battle

yup, only battling in the lost grove counts for rank ups.

The only problem I see with this is it will take 20k uses to level up the basic attack once, unless you gain proficiency at 25%

well, it does take a lot of time, yeah.
You get exp both per usage of the move and for all attacks whether used or not for beating an enemy. The different moves get different amounts of exp for the factors of
> what move it is,
> what rank it is,
>was the enemy regular, elite, or boss.

(The fourth techniques (like decimate, conflagrate) get a lot more for a use/victory than the base attack, for example.)

(And you use the base attack a lot, but. Yeahhh, still a grind, especially with the finishers with longer cooldowns)

Percentage increases for each use, and also for each enemy defeated. Stronger enemies (like named), give a larger boost! (So not necessarily 20k!)

It is 20k uses, you get .005% per use in basic move xp. That calculates out to 100%=20k uses

On top of per use bonus, you also get a bonus for defeating enemies. Defeating named (in the lost grove) gives a larger rank up bonus to all skills.