Job level capped at 60

Just recently, my bard got to level 60, and now the experience bar has a cancel symbol and I'm not getting any exp. I've read elsewhere that the entry subscription that I got with the game caps you at lvl 60, but I've since paid for a standard subscription (a couple days ago) and it still doesn't let me get any exp. There's still a day or two until the 30 day mark from getting the game, but the Mog station says that I have a standard subscription. Do I just need to wait for the next few days or is there something I'm missing?

The entry subscription only comes with the first two expansions (ARR and Heavensward) and 30 days of game time.
If your jobs don't level beyond lvl 60 it means you don't have the Stormblood, Shadowbringers and Endwalker expansion which are required to raise the level cap on your end. It doesn't matter that you're paying the regular sub because you're still missing the necessary expansions.
What you need to do is buy Endwalker on their online store. You only need to buy Endwalker because whenever you buy the newest expansion it also comes with all the previous expansions (minus ARR and HW).

Thanks! It was kind of driving me crazy, so I'm a little relieved that it's not some bug.

There is the free trial, which caps you at level 60 and its associated story and content.

There is the Starter Edition, which removes the free trial restrictions (but doesn't change level cap), gives you your first subscription and costs almost the same as a subscription.

There is the Endwalker expansion, which gives you access to all the expansions up to level 90.

There is the Complete Edition, which is the same as buying Starter + Endwalker.

Then there is a monthly subscription, which is what you need to setup 1 month after buying the game if you want to continue playing.

You most likely bought the Starter Edition. If you bought anything else, such as Endwalker, then you need to register the code to your account so that the game will know you can level to 90.