Gotta be honest, Winhill isn't the most exciting of places now available to you. In fact, well... there's a reason it was left for the end, being perhaps the least interesting bit of optional exploration available to you. That doesn't mean there's not gains to be had and quests to be done, but if you're expecting significant boosts, you'll be disappointed. Winhill can be found along the southern end of the Galbadian continent, just northwest (around a long, barren, rocky island) of Edea's house.
Note: In the plains outside of Winhill you may encounter the UFO again, this time carrying away a cow. Gotta hit all them cliches, eh? Anyways, hopefully you encountered the UFO in the Kashkabald Desert, and if so, go out of your way to run around and find it here, too. Get both encounters and you should have made good progress on this little side quest.
Approach Winhill and note there are two entrances to this town - if you enter via the eastern entrance you'll find yourself in the town square, while if you enter via the western entrance you'll be near the item shop. Use the eastern entrance and head into the large house at the top of the town square screen, then talk to a rather... nondescript man wearing beige/brown who is standing on some stairs. He'll whine about some heirloom vase and a ghost, which is all the prodding you'll get for this quest.
Time to hunt down this vase! You should know the layout of Winhill well enough, so let's laser-focus on what, exactly, needs to be done:
Make sure you have Quistis or Irvine in your party and search a suit of armor along the left side of the mansion. After some chatter, move away from it and a rather odd critter will flee from the dismantled armor, leaving a vase piece in its wake.
Exit the mansion and enter the lower of the two houses to the left - you should recognize it as Raine's bar from Laguna's dream - and head upstairs to get yelled at by a blonde woman. Talk to her until she mentions some flowers downstairs, then head back down and search a table along the lower-right of the screen with some white flowers on it. This should make an apparition of Raine appear behind the bar, which will turn into a cat when you talk to it. Interact with the cat to get a second vase piece. Either this town is haunted, or Ellone's dream-sendings have messed with your mind in a bad way.
Return to the town square and exit via the bridge at the bottom, then in the next screen enter a house to the left (at a three-way intersection in the road) where an old lady lives with her flowers. Search some white flowers at the bottom of the screen. The old lady will ask if you like flowers - respond with Yes and she'll babble about Raine, then search the same flowers again to find another vase piece.
Exit the old lady's house and exit down to the next screen to find some chicobos running about. Standing in the intersection through which they run and interact with one to send it flying. Err... is that some animal cruelty or...? Whatever the case, it should drop the final vase piece. Chase down another chicobo and you'll be rewarded with some Gysahl Greens, then two more to score a Phoenix Down. Nice. Abuse too many chicobos and they'll become wary of you, fleeing at your approach... and a mama chocobo will get some vengeance! Yikes.
Return back to the man in the mansion and give him his broken vase to receive... a Holy Stone. That's right. Just one. That makes this little vase quest about 1/10th as profitable as winning a Krysta Card would be. Oh well, at least it's done.
Triple Triad Rules - Centra/Winhill
Speaking of cards, before you leave Winhill you may want to consider manipulating their local rules. There won't be an immediate payoff, but if you soften the rules in Winhill/Centra now, you'll have a much easier time winning a rare card later.
Winhill and Centra have the same rules, Random and Same/Plus. Ew. You should understand how this works by now - challenge a player in Winhill to a game of cards while you have rules they don't have (Open should suffice, but if you kept the Sudden Death rule from Fisherman's Horizon, that's a good one to use, too). If you do this they should offer to mix rules - opt to play a game with them, and when you get into the Triple Triad screen but before you actually start a match, quit and see if any rules spread or were abolished. If so, and it was something to your liking, exit to the world map and save, if not, reset and try again. If you keep getting unwanted results, try a different player, or play a few games of cards with anybody else and try again.
In this particular case you'll want to enter Winhill via its western gate to arrive near the item shop, where you'll find... two garden students? Talk to them to learn they're dropouts who are nonetheless employed as mercenaries defending the town (can't be any worse than Laguna), then challenge the lower one (the one nearer the exit to the world map) to a game of cards. At the very least, try to abolish the Random rule and spread Open, as Same/Plus can be endured, even if they're not ideal. Once that's done you should be done in Winhill, and hence, done with all the side content available to you at the moment. If that's the case and you're ready for the showdown with Edea, continue on to the War of the Gardens section of the guide.