Thank you for playing the game! To trade, select the item you want to trade for in the other person's bag by clicking on it. Then, make an offer by placing your items in the field between the two bags. The NPC will evaluate the offer. If they like the offer, then they will move their item into the field. Then you can take the NPC's item and they will take your item.
The character who tells you exchange rates doesn't actually exchange anything. That character just tells you the SM value of different kinds of shrooms. For example, if another character says they want 3 SM for something, that means you can trade 3 Basic Shrooms for it, or 1 Tiny Shroom for it.
If you want to practice trading, try trading some Basic Shrooms for items that cost a small amount of SM. The other characters in the center of the Citey have items like that.