[T77911S]

breakers are for wires. wires are sized(breakers) for the load.

you can put a 15amp brkr on 10ga wire. you can NOT put a 30 amp brkr on 14ga wire.

I would run 6/3 to the barn. that will also give you 120. I would also run a 240 compressor. I prefer 240 for motors.

I dint know the size of your barn or what you will be doing but I would really consider a 100amp service out there. when I added my garage I put in a 100 amp box, also converted my overhead power to under ground which meant I had to add a disconnect outside for code. not a big deal. gave me a place to run the garage service from.
don't buy the wire from a box store like lowes. goto your electrical supply house. you can get the wire in the exact length you need and it will be cheaper. I just bought 33ft of 14/2 for less than$10.

I just put in an HVAC in my garage so you never know how much power you need.
compressor,
welder