I'm looking for some kind of measurement equipment , which will be able to read  thickness in few different places(not only edges) of a part. Something what can be used by operator.  Thank you.

a good pair of verniers and a micrometer are usually enough, you get specialised measuring calipers for checking material thickness thats used in the textile industry if you want to measure fabrics.

For example: Let say that the part is 1,5m long and 1,5m wide and has complicated shape.