Will closing vents in closet and bathrooms help cool other areas better?
Yes, this will work, but it could be hard on the furnace blower motor if you close too many vents.

I actually do this in my tri-level home - during the summer I close some of the vents on the lower levels so more cool air makes it upstairs, and in the winter I close vents on the upper levels so the lower levels aren't as cold.