Build a small closet for a 8×10 bedroom
I'd do cabinets like premade cabinets. Takes up less space. - No studs or drywall. Just add rods. Paint to match trim.
Have you considered using a PAX wardrobe from IKEA? It would be easier...
Premade cabinets are easier, but look for those designed as wardrobes. Ikea, for one, has many inexpensive models like this one: https://www.ikea.com/us/en/p/kleppstad-wardrobe-with-3-doors-white-20441757/ Remember that ALL cabinets and chests in children's rooms should be attached to the wall to avoid accidental tip over.
If you want a closet for resale value, you are better off building one. Most cities in the U.S. require a closet in order to advertise a space as a bedroom. Otherwise the room can only be listed as office, hobby, or storage space. Here is an article on building: https://www.hometips.com/diy-how-to/closet-framing-build.html Remember to remove existing trim before attaching studs to the walls as the trim will prevent the studs from fitting against the walls. You can measure and cut the trim to fit against the closet (inside and out) once you are finished.
Before deciding where you will place the closet consider where the door is and how it opens. You need to leave room for a door to open. Also consider whether windows might be in the way. Clothes need a minimum of 24" interior depth for hanging. If there is less than 30" between a window's trim/edge and a wall corner you may need to put the closet on another wall. IMO, the closet is best on a short wall. Click on the picture below to see three possible layouts: first one retains most open, usable footage; second retains open, usable footage, but closet space is smaller; third has one window in the way, so closet must be placed along one long wall.
I can't tell you how to build the closet, but I will suggest that you consider using a "Barn door" instead of a hinged door because of the limited space you have in the room. Or you can use bi-fold doors. I have bi-fold doors on my closet because it allows air circulation to eliminate the musty smell of closets.