Nook's Cranny is where you should sell/trade. There are some guides on how to start your own shops also that you can look at.
The Airport is a good place to host giveaways for items, let people know if a specific NPC is visiting, if you want to give an island tour, share turnips prices, let people know of meteor showers, etc. (basically things that require the in game airport, but do not count as trades or buying/selling).
An it looks like you already found the
New Neighbor Network which you can search for villagers you want or put villager up for sale, trade, giveaway.
All of these places should have guides associated with them, but if you need more info there are people that can help.
And yes, that's basically what you'd do. You'd list what items you want to trade or sell and when someone responds you can set up a time and place to do the transaction.
