Buying and selling furniture happens in the
Re-Tail forums, and villager trading, buying, giving away, etc. is for the
Villager Trading Plaza.
Typically, if you buy furniture from someone here, you will decide on all the details on the forum - things like what furniture you're buying, the price that works for both of you, who is dropping off/picking up, exchanging FCs. Then, after you've both added each other one of you will open your gate and make the trade. Depending on what you decided on, you'll either just trade by the train station - people commonly exchange items by dropping them on the ground near the train station. Sometimes people like to put it up in Re-Tail for the decided price, and the other trader will just buy it from the flea market there.
If you want to give away a villager, you should create a thread in the Villager Trading Plaza and state which villager you're giving away, whether they are in boxes already or not, etc. Some people get creative with giveaways and make it like a contest, asking people to draw things, stuff like that. You can take a look at how some people set up their giveaway threads
here.
One thing to remember about giveaways though, if you don't like that particular villager, there's a chance other people don't too - you can check the villager's "popularity"
here. It's not 100% accurate, mind you, but it's a good guide to who sells for a lot and such (for some villagers, people are willing to pay millions of bells for).
For a villager to actually leave your town and be "adopted" by a certain person, you have to have that villager in boxes (already packing up to leave), then open your gates, let the chosen person in to talk to that villager, and they will move out of your town and into their's the next day.
Hope that helped, let me know if something's still not clear. c: