I'm trying to move around some of my sylvesties just to organize them into pairs or whatnot. I'm not seeing an obvious 'move pet' button, but I could very well be missing it. Anyone know if I'm just leading myself into a dead end lol.

To move the Sylesties in your stables, there's two ways you can do so:

The first way is by clicking your Sylesti(Don't click their name though), then clicking the tab you want to move it to. After doing so, you'd click the stall you want to place the Sylesti in.
If you're trying to move the Sylesti to a new stable, you'd click the Sylesti, then click your stables drop-down menu(It's to the right of your avatar image, underneath Occupancy). Click the stable you want to move the Sylesti to. Sylestia will load that stable for you, then you can click the tab you want to put the Sylesti in, and finally the stall.
Your Sylesti will be moved to the stall you selected.

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To mass move Sylesties(Multiple Sylesties moved at a time), you'd press the green Enable Mass Move button.
After doing this, you'd click each Sylesti you want to move. You can click the Sylestis in the order you want.
After clicking each Sylesti you want to be moved, next to the green Mass Move button that'll have appeared(This will be in your stable), you'd have two drop-down menus. One for your stables, one for your stable tabs. Select the stable you want the Syesties moved to and the tab in that stable you want them moved to(And yes, you can use this feature to move your Sylesties to a different tab in the stable they're already in).
After that, click the previously mentioned green Mass Move button. They'll have then been moved to the stable and tab you chose.
If you want to further organize them, you can use the first-mentioned way to do so.

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I hope this explanation helps!

Ahhhhhhh, that makes sense, I kept clicking their names and not the 'stall' the pet occupied, thank you so much!