Mobil-1 is the factory-fill oil for Corvettes, although you can use any 100 percent 5w30 synthetic oil you like. My preference is Royal Purple, which I've used in this car for several years now. I am also a fan of K&N oil filters (I especially like the built-in nub on the bottom of the filter which makes removing and tightening a breeze), but this, too, is a matter of personal preference.
Which brings us right to the point: what's the right way to change the oil in your C5 or C6? Here's how to go about it.
In the way of tools, you're going to need a 15mm box wrench or shallow socket and ratchet, a drain basin capable of holding up to seven quarts of oil, and a suitable container to hold the spent oil for transport to an environmentally correct disposal facility. (Auto Zone and many other automotive parts retailers will gladly accept your used oil for recycling or responsible ecological disposal at no charge.) You'll also need a strap oil filter wrench unless you're using a K&N oil filter, in which case you'll need a 1-inch open-end wrench or 1-inch socket and ratchet. It's also a good idea to have some paper towels and newspaper available to catch and soak up any stray oil that the drain basin doesn't catch. I'm also a big fan of nitrile gloves to keep the grimy old oil off my mitts.
The first thing you'll have to address is elevating your C5 or C6 and supporting it while it's off the ground. A pair of gradual-incline ramps such as the Rhino Ramps offered by Mid America Motorworks is ideal since they're designed specifically for low ground-clearance cars like C5/C6 Corvettes. These ramps are made of strong, lightweight plastic and the bottom surface is waffled to distribute the weight evenly across the entire contact surface of the ramp. What this means, simply, is that they won't dig into the tarmac of your driveway or scratch up your garage floor the way pressed-steel ramps do.
If you don't have gradual-incline ramps like the Rhino units, you can elevate the front of the Corvette using a hydraulic floor jack, also known as a trolley jack. Bear in mind, however, that the bodies of C5s and C6s curl and wrap around the hydro-formed framerails, so it's easy to dig into and crack the body plastic on the underside of the car. The framerails, however, have oblong/oval shaped openings at the front and rear on both sides of the frame, and these openings are intended to be used as jacking points. You can purchase a jack insert that will replace the stock lifting platen of your trolley jack, and this insert will fit into the C5/C6 openings like a hand in a glove, thus permitting you to lift the Corvette properly without causing any damage. Another option is to purchase a set of four aluminum lifting pads that fit these openings and give you a solid platform to jack from. A third alternative is to purchase a set of four plastic lifting plates that serve the same purpose as the aluminum pads. The jack insert, the lifting pads, and the lifting plates are all available from Mid America Motorworks as well.
Or, you can also use your trolley jack to lift the entire rear end of the C5 off the ground by placing it under the transaxle crossmember. If you go this route, you'll need a trolley jack that has a very low profile, and I strongly suggest using a padded jacking platform such as the one offered by The Eastwood Company. This padded platform simply drops in place on the trolley jack just like the stock lifting platform that came with the jack.
Regardless of which of these methods and mechanisms you use to elevate your C5, you're going to need a pair (or two pair if you don't use ramps) of jackstands to support the vehicle safely when it is off the ground. It's time for me to get up on the soapbox now, so bear with me while I do a little preaching here. Never, ever, get under your C5 or C6 (or any car) for any reason without supporting it properly. Jackstands are inexpensive insurance against injury to you and your beloved Corvette, so don't be foolish to work on your elevated vehicle without having them in place. Nothing can ruin your day like having your Corvette come crashing down on top of you, so don't be penny wise and dollar foolish when it comes to equipment. Enough said on safety, so let's move on.