On the surface, thickness seems like an easy measurement to take, as dimensional measurements go. You have two big, flat, parallel planes on which to plant your gage. For hard materials, like metal sheet, you can use three of the four measuring capabilities of a slide caliper. A micrometer does very nicely, too, if you have the time and need the accuracy. If the need is constant, then speed becomes a consideration. In this case, dedicated thickness gages exist, which are fast, simple, and spring-loaded to make the measuring force repeatable.
Non-flat materials can be checked with much the same methods. All this requires is a spherical or conical contact point for your gage. Dedicated gages are available, as are add-ons for most common gages.