The Gamo is a nice rifle and for your needs it fits the bill and can easily be picked up at a one of the Sporting outlets. My first air rifle was a Crosman I bought at Walmart. It really got the bug going. However, if you ever decide to continue the hobby, it would be wise to at least try out one of the Beemans. ONLY BEEMAN R SERIES. These rifles have superior craftsmanship and and the triggers are exceptional. They are designed to last a life time. The Spring has less vibration and twang and better seals.
One of the most popular is the Beeman R7. Beeman R series are all actually Weilrauchs imported years ago by DR. Beaman a legend in the air gun world. The R7 is a light weight rifle that will be a great trainer for younger shooters and a gun that they can grow into as a adult.
http://www.beeman.com/r7.htm
http://www.straightshooters.com/beeman-r7-.177-beech-elite-combo.html
Want a Super tuned R7. I had this guy tune mine with a special kit that increased velocity, smoothed the gun out to the point of ridiculous and took a already quiet rifle into and practically made it silent. Yes, you would pay more this rifle, but well worth the cost for a lifetime rifle.
http://vortekproducts.com/ourstore/Airgun
Probably the most popular starting rifle among the Airgun guys, (my first move up", and another German made quality is the RWS Diana MDL 34. If you have a little bit of skill, you can get a vortek Kit and install it yourself.
http://www.pyramydair.com/s/m/Diana_RWS_34_Breakbarrel_Rifle_T06_Trigger/402
Glad to hear your comments Slowhand. Good luck with the tree rats. They are serious destructive pest. If you get a chance, nail a few Starlings. They are killing the American Song birds. Right now, I am on the hunt at night for a very elusive RAT that hits my Bird feeders at night. He is a wise old guy, and alert, but I will get him soon.
Anyone that has questions about airguns, you are welcome to join the "Yellow Forum".
http://www.network54.com/Forum/79537/
They have subcategories for all airguns including Gamo and Crosman etc. Also they have a "Classified section" where you might find a great starter rifle.