You are right, you want Access Points.
Ubiquity seems popular, but I will let others chime in.
Use single mode fiber. Ideally put it in a conduit. Run more than one cable.
Ethernet over the single mode fiber. Distance is not an issue, you can go miles over single mode fiber.
Hard wiring with ethernet is the best option. Dig a trench and run single mode fiber between the house and barn.