Looks pretty simple with no tile destruction. At the end of the stem it looks like there is a "nut". Unscrew this. It is the packing nut. after you remove it, you should be able to turn the stem and unscrew it out. On the end of the stem you will see a rubber seal/washer held on with a brass screw. Take the whole stem assembly with you to a hardware or Home Center to get the right size washer. Buying a pack with multiple (different size) washers is a waste of money since you will probably never use the rest of them. Remove the screw, replace the washer and reassemble the stem. The wiggling may have been caused by the worn washers. If not, then you need to get new seats and install them. Try it with just the washer replacement first.
if you have never made a mistake, you probably haven't done much.