Do some searching on the internet about turning green wood. You will find a lot of information to consider. some of it good and some of it pure speculation.
Here's what happens. When a tree is cut it is full of water. As it starts to dry it drys faster by the water leaving the ends and the outside drys first. When wood drys it shrinks. so the outside is drying and shrinking and the inside is drying slower and staying about the same size. Imagine two circles one inside the other. The outside one is getting smaller and the inside one is staying the same. Something has to give so the wood cracks.
Wood takes about a year per inch to dry so a tree that has been down for a few years is still very green on the inside. In fact I read the other day that wood that is down will tend to stabilize after a while and take much longer than a year per inch to dry.
How do you deal with this. Well you have to either turn it thin so it can dry and warp without cracking or you have to slow down the drying time. It gets to be complicated about how to deal with this because every project is different. To start off with split the wood through the heart so you've brocken the circle. This lets the wood move a little and lessens the chance of checking. Then seal the end grain with a sealer. I use either wax or Anchorseal end grain sealer. This only slows down the loss of water but won't stop it from checking if it drys too fast.
to get around this I try to harvest the wood. I saw it into 2x2, 3x3, 4x4 etc and seal the ends. Then I put it up off the ground to dry. It will take a year or more but if you do this often enough you will always have some dry wood to deal with. Thing like candlesticks or lamps that are fairly thick have to be completely dry or they will crack when you take them inside where it's really dry and warm. To do boxes I take the 3x3 blanks and cut them down to 4 or 5 inches depending on the length of the box and then put them in the microwave. I heat them up until they are warm and let it cool and do that again. It may take a dozen or more heatings to dry and it still may split but I've had pretty good luck with wood that has been sitting for a while before being subjected to the microwave.
Short candlesticks might also be dryed in the microwave. The reason this works is the microwave heats from the inside out and equalizes the drying stresses. It can smell up the microwave so be careful or buy a cheap one for the shop.
To turn large lamps I hollow out the inside so the sides of the lamp are only 3/8" or less. this lets it dry more evenly although it will warp depending on how the grain runs through the piece and if there is a knot in the piece. it still might check at the knot.
To turn bowls as soon as possible I rough out the bowl so the walls are about 10 percent as thick as the bowl is large. I seal all endgrain areas and then put the bowl in a paper sack or box to restrict the air movement. You want even wall thicknesses all the way around, even the bottom. I let this dry until it stops losing weight and then re-turn it for the final product. It takes about 6 months for the average bowl.
to turn a bowl to completion in one sitting I turn it so the sides and foot are all about 3/8" or less. If it takes a while to turn the bowl I spray the outside with water to keep the whole piece drying at the same rate. When I finish the inside I lightly hit the whole piece with a hair dryier to dry the surface. Then I sand the bowl and put it in a paper sack to dry for a few days. It will warp so when I pull it out I sand the foot flat and it's ready to go.
OK that was a long winded scenario of how to deal with green wood and I just touched the surface. Read all you can. Or you can do what I did early in my turning career and that is to glue up dry wood to get the sizes you need. There are some excellent books on segmented turning and since you have a shopsmith you have all you need to do that. I started with a shopsmith. Order Malcolm Tibbets book on segmented turning and you will learn all you need to know to do accurate pieces.
Amazon.com: The Art of Segmented Wood Turning: A Step-by-Step Guide: Malcolm Tibbetts: Books