First you need to check with you county/city and see if you need a building permit for your renovations. This protects you if you ever need to file an insurance claim, and also when the time comes to sell the property.
That depends on what you will be doing with the barn. If you'll be housing animals then you need to think first about how to deal with their wastes and getting them clean fresh air, and pick products that are appropriate for that. (ie cellulose insulation is probably not the best choice)
If it's for a shop, then you have many of the same issues you would have for a house, but you will have higher ventilation needs. but you may want to pick products that are more water tolerable, depending on usage.
Since you are talking about redoing the entirety of the siding the cheapest option is likely putting in appropriate sheer panels, then sheathing with foam board insulation then vinyl siding. One advantage of vinyl siding is it has the air gap built in. If you want to reuse the siding, then you will need to find some new siding to mix in with the old, and you will likely also need to mill the old siding to get rid of gaps. Then you can follow Hammer and Hand's detailing for vertical siding:
https://hammerandhand.com/best-pract...rtical-siding/ an alternative is to place the furring strips on a 45º angle. then you don't need the horizontal strips. Doing it that way you will want to ensure there are gaps in the furring strip lines so that the gaps can communicate with each other. Matt Risinger shows this detail in his metal roofing:
The best way is to sheath the barn in plywood, apply a fluid applied membrane to the exterior side, apply a rigid mineral wool or fiberglass board to the thickness required, then hammer and hand's furring and siding detail. Because Mineral wool and fiberglass are so vapor open and don't absorb moisture they lead to the best drying capacity of all systems. Don't forget to prime all 6 sides of the siding.