Dave, I don't know your specific questions or problems so I'll just note some things. Ask if you want to get more specific.
Rent a power miter saw. If you really can't find one for rent, make a sled to ride in both tracks of table saw slots and with an exact 90 degree center section sawed with a slot to exactly bisect the angle. Don't even think about hand sawing, even with a miter box, unless you are very experienced or want to scrap a lot of moldings.
Do your measuring on the molding and from the space to fill. Trying to work with a measuring tape is too inaccurate for this unless you are willing to cut and recut to get close to perfection. When you do get really close, you will not be the first one to add a little white caulk if you are going to paint the moldings. If you are going to have a stain finish, prefinish totally, cut-to-length and nail in place, then fill the nail holes with the Minwax Wood Filler that is colored wax. Use a nail set and "smiley faces" from a hammer don't look professional.
On long runs, like base molding, I usually cut a scoochie bit long and nail into exact placement. Miter outside corners and cope inside corners of base, crown and chair rail moldings.
Make sure that the square you are using is really square. Checking it against a couple of "factory corners" of MDF or MDO and finding any discrepancy is a reason to check further to be sure that your square is square. Using a cheap combination square will cost you a lot of work, molding and money. Don't just mark length--put a rough line on the unfinished side to show which direction miter cuts are supposed to go.