I turn the key and it just cranks real slow. After a couple seconds it will speed up just enough to start. Brand new battery, 2 months old and fully charged when I try to start it. Battery cables, mini starter, and harnesses are about 4 years old. About 2,000 miles on the car in those 4 years. One day it was dead at the ignition but I was able to get it to cranking (cranked very slow but enough to start)from the relay box. I did check the negative battery cable connection to the block and it appeard to be good. Headlights are bright when turned on but do dim quite a bit when trying to start the car. What can I do to solve this?

Hard to say. Just because the battery is new does not mean it's big enough, and "cranking amps" are easy to manipulate by the battery company. Is it a good, solid brand name, and what IS the rating?

Could be starter, battery, cables, or even a problem with the engine.

First thing I'd do, is find someone with a good old fashioned load tester--known as a "carbon pile" and find out what your battery will put out, and what the starter draws.

The cables could be marginal--everything nowadays is subject to "poor quality"

You are asking "what's wrong" and the answer is, "it could be most anything."

Its a 700 cca battery that I picked up brand new at interstate battery.