Look for straight sections of the trunks or limbs. Check the bark for any signs of twist. I have noticed that osage trees that grow near sources of water have better rings. Seal the ends as soon as you cut them. If you can get the logs out without splitting them, go ahead. You can always split them at home. I usually cut them into lengths of 75"-80". That allows extra for checks on the ends. It also gives you some wiggle room if there are any hidden knots or other bad areas that you might have to design around. Keep an eye out for any "Character pieces". After you get them out, you have several options. Some people split the logs in half or quarters and let them dry that way. Some people go ahead and split them into staves and leave the bark and sapwood on them. If you do that, make sure to spray them with some insect repeller. I split them into staves and go ahead and remove the bark and sapwood. They take up less storage room that way and are cleaner. Plus when you are ready to build a bow, you can see the staves better and pick out the best one for what you are building. If you remove the bark and sapwood, seal the backs as soon as you can. Maybe try some each way and find out what works best for you. Good luck and let us know what you get.
