Your catalog will be upgraded.
If you open it with a later LR, it will make a copy and upgrade the copy to keep your old catalog save.
It should work. LR CC will convert a copy of the LR2 catalog so you’ll still have a LR2 catalog that will open in LR2.
Here is a page that describes the process of moving LR to a new computer:
I’d probably only copy the .lrcat file and leave the Previews.lrdata folder so LR computes new previews as needed.
If you are jumping from LR2 to LR6/2015CC without any acquaintance with 3 and especially 4 or 5, you had better set yourself for a huge case of future shock. Fundamental changes and improvements have been made in the Raw conversion engine (LR3) and the entire approach to tonality editing (LR4), several new ancillary editing tools added and old ones improved, and multiple UI changes may seem puzzling.
I would recommend Free Lightroom Quick Start eBooks