As with any building, the barn's foundation is all-important. Many old barns have stone foundations, some with what's called dry-laid stone. This means there is no mortar to bind the stones together. The lack of mortar in and of itself doesn't mean that the foundation is not sound. In fact, early stonemasons were highly skilled and built many enduring stone structures without mortar. Whatever the construction of the barn's foundation, inspect it carefully. Especially look for bulges or slumps in a stone foundation and cracks or other irregularities in a concrete or stone and mortar foundation. Any compromise in the foundation must be addressed before any new wooden structure is placed upon it. In other words, work from the bottom up.
The two major parts of the barn's wooden structure are the framing and the exterior. Consider these separately and, just as with the foundation, work from the bottom-up. One key part of a barn's framing is the so-called sill. This is the part of the barn's wooden structure that rests upon the foundation. It is, essentially, the wooden foundation of the rest of the barn's wood structure. Because the sill is close to the ground and in contact with moisture, it tends to be more subject to rot than the rest of the barn. It is imperative to ensure that the sill is free of rot and that it is otherwise sound.
Once the foundation is deemed sound, shift your focus to the rest of the structure, still working from the bottom up. Many old barns were so-called post and beam or timber frame construction. Unlike modern construction using dimensional lumber that was cut in sawmills and assembled with metal nails, many old barns were made with hand-hewn posts and beams fastened together with wooden pegs called trunnels. This method of construction---just as with the old masonry---was very sound and enduring. Still, as with any structure it does tend to deteriorate over time. Insects, like carpenter ants or powder post beetles, can infiltrate the wood and weaken it. Inspect the framework from the bottom up and replace any failing members.
After the foundation and framing are addressed, work from the inside out. The barn siding and roof boards and shingles are like flesh and skin on a skeleton. Identify and replace any defective siding and roofing as needed. Some original roofs on old barns may have been made of wooden shingles. Unless there's a compelling architectural reason to retain the original character, you might consider a modern asphalt shingle roof or even a metal roof as a replacement for a barn.
In summary, even though the barn is composed of separate components, think of it as an integrated whole because one part does affect another part and ultimately they all function together as a cohesive unit.