I am installing a new 200A load center (main) for a 100A (existing) panel. I noticed that the ground wire is just a 14/2 wire run outside:). I will need to run a ground from the new panel but the water meter is about 25' away. Can I run the new bare copper ground wire through the wall and into the ground outside? What size wire should it be and would a grounding rod be easier to put in or would it be better to use a grounding plate

Since you have city water, and I will assume that is is metal pipe, you must run a #4 copper to the point the water line enters the building, jump out the water meter with the same wire, using two pipe clamps for #4 wire. You then must install either a ground rod or plate and either continue with the #4 or run a #6 from the neutral/ground bar in the Main to the rod or plate.

I have only used rods, more readily available.