The auctioneer starts doing his thing the second you walk in. After the screen fully loads, he'll call out a list of items for sail. To buy an item, walk up to the stage until a "?" appears over Zidane's head, then press X. A bubble will pop up, where you can make your bid.
The following table gives some suggestions of prices at which you should stop bidding in order to save money:
Rare and Key Items that you can bid on at the auctions include the following: Mini-Cid, Griffin's Heart, Doga's Artifact, Une's Mirror, Magical Fingertip and Rat Tail. If you're a FF veteran, these item names may seem familiar. See our classics connection page for more details.
One thing to always keep in mind while at the auction is that the Treno Nobles have a ridiculous amount of gil, and will put it to good use bidding back and forth. Sometimes, the bidding will go on for a pretty long time. In order to almost always win an item, wait until it seems that the item is about to be sold, then place a bid slightly higher than the last.
Some from the auction can also be sold to nobles! If you've bought any of the items listed below, take them around to the Café Carta (outside and inside the building that has the synth shop) and try and sell them to some of the nobles. They're bound to give you some great prices! Talking to some of them twice may even reward you with some extra gil!
Note: You must sell the Griffin's Heart to the Adventurer before he becomes interested in the Rat Tail. Also, the Scholar will not top his original offer of 10,000 gil.
Dark Matter: Can be used in battle (inflicts 9999 damage on enemies) and can be used to learn Dagger's Odin summon.
Doga's Artifact & Une's Mirror: Go to the Black Mage Village Inn and check the gramophone. By examining it, certain arranged versions of Final Fantasy 3 music will play. Note that this even is only available on disc 4.
Magical Fingertip: Give it to the old man in Daguerro Library, who will reward you with Steiner's Excalibur sword. Note that the Magical Fingertip will not appear until you have bought Une's Mirror, Griffon's Heart, Rat Tail and Doga's Artifact and sold them to the nobles.