Hi, John,
Looks like you're off to a good start! If Gamo hasn't changed things, the scope is a BSA 4x32. This is an okay scope, but the gun is capable of utilizing a lot more scope. And it's not an AO (adjustable objective) model, so you can't "dial it in" for different ranges, plus parallax can become an issue. So I'd consider a better scope to go along with your GRT III trigger. Regardless of what you do scope wise, GET GOOD MOUNTS! I've never been impressed by the mounts Gamo supplies with their guns. Accushot by Leapers are good mounts at decent prices, around $15. From there, you jump to the $50 range. That gets you BKL's and the Beeman and B-Square adjustable mounts. Any of these are far superior to what Gamo supplies. Oh, and if you do decide to upgrade your scope, make sure you get one that's rated for springers.
Regarding pellets, chances are that a domed pellet in the 8-8.5 gr. range will be the best in your gun. JSB Exacts and RWS Superdomes are two very good ones in this range that work well in a lot of guns. Beeman FTS's are a little heavier, but work well in lots of guns, also. Any of these would be a good general purpose pellet for ranges out to 50 yards or more (depending on how good the shooter is :-)!). And a lot of guys like the Crosman Premiers. They're a little lighter at 7.9 gr. but work well for some guns. If you plan on doing any hunting, any of these will work, but try the JSB Predator. If your particular rifle shoots them well, they are a devastating choice for small game, with very good range and retained power. For shorter range hunting, say out to 30 yards, Beeman Crow Magnum HP's can be good. For shorter range paper punching, out to 20 yards or so, I've had good luck with JSB Match Diabolo S100's. They're a wadcutter design, and are very accurate in my CFX and bosses Varmint Stalker (older version of yours). Gamo Match are okay, but most of the Gamo pellets are pretty variable in weight. Crosman Premier Match pellets work well in one of my guns. Beeman and RWS both have match style pellets in various weights. You can also order pellet samplers. I'd get the one from Straight Shooters. Lots of variety at a decent price.
The only other suggestion is, shoot, shoot, shoot! These things are a blast. Just be aware, they ARE addictive! Your wife may come to regret this particular Christmas gift :-)! After you've had if for awhile and shot it a lot, then you can have it tuned. You could do it right away, but you won't appreciate it nearly as much. Oh, and learn the proper technique for shooting a springer. Look around, the info is readily available. Good luck, welcome aboard, and keep us posted on your progress.
Dave