I think it's important to stay with a FULL synthetic, as some oils are a mix of dino oil and synthetic. The other very important point to consider is that the oil meets or exceeds GM4718 - which is the standard for the LS1/LS6 engine.
Mobil introduced a new line of Mobil 1 oils called extended life - for 7500, 15,000 miles - etc. These oils do not meet the GM4718M standard and I would not use them. I understand that Quaker State and Pennzoil both are now offering full synthetics - and that they do meet the standard. If the oil does meet it - it will say it on the bottle, so look for that.
As far as filters go - I personally prefer the UPF44 (Delco Unltra-Guard), which are generally not sold in the stores, but are usually available from dealers and Corvette Specialty places. There are others that are good - including the Mobil 1 filter.
As to when to change the oil - you do realize that the Driver Information Center has an oil life monitor, right? The problem is if you just bought the car, an unscrupulous seller could reset the monitor without changing the oil - indicating that you have 95% oil life remaining. If you change it, make sure you reset the monitor - then I would follow that and change it when it tells you to.
GM spent a considerable amount of time developing that monitor, and with the full synthetic oils - you might only have to change the oil once a year or so, depending on mileage and use. The monitor is not optical - meaning it does not look through the oil to determine it's clarity or anything like that. It uses a complex set of algorithims based on throttle position, rpms's, speed, etc. to determine how the engine is run. An engine that is run hard will use up oil life faster than one that is driven gently.
Hope this helps - and welcome to the C5!