To give items in Story of Seasons: Trio of Towns, you need to have the item ready. Locate the intended recipient and get within their interaction range. An option to give the item should then show up. Keep in mind that some items might be more appreciated by certain characters than others, influencing their response.
First, make sure you have the item you want to give in your inventory. Then, find the person you wish to give it to and approach them. When you get close enough, a prompt should appear allowing you to choose the option to give the item. Different characters might have different reactions depending on the item and your relationship with them.