A lot of good questions, and while there are general answers, there are always exceptions.
First, there is a lot of BLM and other public land which is surrounded by private land without any access right of way. Sad, but true.
If there is a public roadway going through the private property (county road), then of course you can use the public road as well. Not always possible to tell on a map. but typically numbered or named roads are publicly maintained and therefore open to use. If public funds are used to maintain a road, it is open to the public. Two track routes are not roads by this definition, and you can not cross private property on a two track without permission.
It is not uncommon in a lot of this area for a county road to go effectively right through the middle of a homestead, much like a driveway. House on one side, barn on the other, and equipment and vehicles scattered about. It can be really tough to tell if these are private driveways, even for those who live in the region.
Best advice for a total newcomer is to look for large blocks of BLM or National Forest with county road access. And by the way, the asterisks are based on a percentage of private land in the unit. Some of these units have very large chunks of public land, while the remainder of the unit is almost entirely private. There is some great public land hunting in some of these units. You might consider buying the WY Atlas and Gazeteer (a map book) for a good starting point. These are about $20, and available at most Sportsman's Warehouse stores and other recreational outlets. You can quickly determine which units have decent public access to large chunks of public land. And when you find the checkerboard private / public areas, you can start on an entirely new learning expereince.