What size breaker and wire for a barn feeder circuit?
Based on the activities you are talking about, I would say #4 aluminum.  

The reason is 40A will probably suffice for immediate concerns, and you can do that with #8 copper except for the distance.  That necessitates a bump to #6 copper, and a side effect of this bump is you get some more headroom, as you can technically drive that circuit as high as 60A.   

However, #4 aluminum has the same ampacity as copper at half the price, so why not use it?  At these large sizes, the fearmongering about aluminum does not apply, and these wires will be the newer alloy anyway.