There's a couple of ways to go about it. Quite often milling or logging is in there family. Those folks usually have a good bit of experience by helping out in the family business. Skills they develop are equipment maintenance, safe felling techniques, safe equipment operations, log bucking, markets, and many others. If you don't have that background to rely on, then you have a few other options.
You could go and get an associate degree in forestry. Usually that involves logging, timber cruising, forest management, and you can even get into equipment maintenance and some business management. I know several guys that went this route and became very good and highly professional loggers. You would have to search out which school meets your expectations, so search around.
If you want to get straight into it, you'll have to find a crew that will take you on and show you the ropes. Most small crews aren't set up to do that unless you can bring some experience with it. If you can run equipment, or good at cutting trees, then you might do okay.
Mills are not as tough to get into. They are always looking for mill hands. I did that during the summer while I was in college. After I graduated, mills were about the only place I could get jobs. Even with a college degree, I started as a mill hand. If you apply yourself, you can learn the ropes of the business pretty quick. But, to get into sawing, edging or any management often requires an expanding company or someone ready to retire. You may find someone that will sponsor you to learn how to grade lumber. It's a 6 month course in TN, but you're certified.
So, if you don't want to go the education route, get a list of loggers and mills from the state forestry dept. Then go knock on some doors and sell yourself.