Oil based will give you a harder enamel-type finish, but if you have any water get to the wood under the paint, it doesn't have anywhere to go, it can cause rot, but more likely makes the paint peel off because it can't adhere to wood that is constantly wet. It cleans up easier, and has other benefits, but latex lets the wood breathe a tiny bit more, so on the outside, it won't have to be touched up as often.
Either way, make sure you get a good mid-grade or better paint--don't buy the cheapest stuff out there or you'll be repainting way too often. I swear by Kwall and have done a ton of painting with it, but I know Kiltz is also a very good brand--they have a killer primer that will totally block out pen and marker (unrelated to chicken coops, since few of my hens have ever draws pictures on the wall). I've never heard anything bad about Sherwin Williams, but I've never used it myself.
One good coat of primer will do you, and use good quality rollers and decent brushes to minimize extra coats and lines in your paint. A good paint will give really decent coverage with one coat, and maybe a light touch up. That's usually all it takes for me, but if you're painting OSB or other wood product, it may take a bit more because it soaks in better than on sheet rock. If you plan to use a bright red or yellow, expect to use many more than two coats. I did seven coats of cranberry on my bedroom wall. It was well worth it for one wall, but I'd hate to do that for a whole shed.
I've been working in a paint store for years, so take this for what it's worth. I hope this is helpful! Happy painting.
