We really need to get a 350Z buyer's guide up here, especially since they're so inexpensive to buy anymore. Anyway...
The '03 & '04 models are known to have problems with the power window motors eventually failing. Also notchy shifting in the 2-3 & 3-4 shifts. If the previous owner had a transmission replaced with the later CD009 transmission this will not be an issue.
The '05 track & 35th Anniversary (6MT*) models had oil consumption issues. They had the RevUp version of the VQ35DE engine and some of those engines had a bad cylinder hone that was the source of the oil consumption. I've got a 35th Anniversary and had no oil consumption until the the mileage was really high.
In 2006 all 6MT models came with the RevUp engine.
In 2007-08 the 350Z got a newer twin intake engine, the VQ35HR along with a bit nicer interior than the older 350s. My suggestion is that if at all possible purchase from these model years only. You'll get a better, stronger engine and a better interior.
Welcome to NICO!
*Six speed manual transmission