Agreed on a stove that sources outside air. That not only removes a lot of risk from fumes igniting, but it will be more efficient since it isn't sucking cold air in the gaps.
Stoves are pretty simple things. box with flaming wood. Most will have a thermostat that controls a thermostatic spring which opens and closes an air door. Duct that air inlet to outside the barn.
Always burn bone-dry hardwood. Never burn soft woods (except maybe some kindling to start). Soft woods have lots of pitch (even when dry) that burns dirty and leaves creosote deposits in the chimney which can burn. This is what causes chimney fires.
Clean the chimney once a year and cap it during the off-season to prevent bats, birds, and other critters from using it as shelter. It only takes one wasp nest, one chimney swift, or one raccoon to really ruin your day. Make sure your chimney has a cap vent (umbrella thing). It's not a huge deal, but water going in your chimney mixing with the creosote not only makes it corrosive, but also turns it into sticky paste instead of easily-removed dust.
My dad heats the house with a wood stove. Every spring when he "retires" it, he drops a weighted brush down the hole several times and shovels out the junk, then he lays a piece of plate steel on top. He is considering an umbrella cap and screen so he doesn't have to use the plate, but then he will have to remove and replace the cap and screen every year to clean it. Pretty easy if you have a stove pipe. Not so easy for him with a masonry chimney.
In his case, having a permanent cover isn't as much a need. He lights the stove in October and it usually burns until March with very few interruptions. If you're just going to be using it every once in a while, a permanent cap might be best. Slide it off the pipe once a year to clean the chimney
The only other thing I can suggest is, when you light it, keep it going for a while. You want to get the chimney good and hot. Kinda like a car exhaust system. If you just do short commutes, the water condenses in the cold exhaust. If you run it long enough to get the tubing hot, it evaporates the water. Same with a chimney. Wood combustion gives off water vapor which condenses in the chimney. Burn it for a few hours to make sure the chimney gets hot enough to evaporate the water