I'm looking at my pets moves and I'm not quiet sure how to level up Slash from rank 1 to 2

https://www.sylestia.com/proficiency/?pet_id=7183820

You just have to wait for a long time. Youll start to see it go from 0% to 100%, but itll take a very long time.

You do need to be battling in the Lost Grove to rank up your moves! If you aren't there yet then you won't see any progress.

Huh, I didnt realize that.

as has been mentioned, it only happens in the lost grove- this is actually in the Help Info on the proficiency page, if you click the ? and scroll all the way to the bottom of the popup:
"... Pets can earn Expertise by Battling Enemies within the Lost Grove, a Level 60+ Zone. Each Pet will have its own Expertise Rating for every Ability in each Element. ..."

Even when you do get to the Grove, it can take a long while (although the item gained for achieving the first rank 9 reward from the Patriotic event increases the gain of rank expertise, as do philters of expedient learning) (the results of ranking speak for themselves though)

You can buy ranked pets on occasion, or pay other people to rank your pets for you, but the prices for these cases are up in the millions (people who have access to, and teams fully set up for, the third section of the grove can get ranking done pretty fast, but it's still a lot of dedication to do)

That makes sense, thank you everybody!