I would recommend picking up some cab, zinfandel, and pinot noir and setting them aside - by the time they are ready to drink, you'll likely be ready to drink them! Ideally, reds should be stored in a cool, dry, dark place (like a cellar); store them on their sides, not upright. They usually are at their best when they are at least 5-6 years old.
For lighter reds, try a grenache (one of the very, very few reds not served at room temp - serve lightly chilled, not cold, though, just cool). Blends are sometimes a little lighter, too.
Cabernet Sauvignon (and its cousin Merlot) is a good starter wine. It goes with a lot of foods, isn't too strong, and is easy to find. Zinfandel is a strong but versatile red, but it can be harder to find (my favorite zin is probably Rancho Zabaco - California and Washington both produce some good zins). Pinot Noir is also very good but strong. It's a type of burgundy. Some of the Oregon pinots are very nice, if they are available over there. I like most reds with beef or highly seasoned foods, or with strong flavored cheeses or other appetizers. I also like zin with waterfowl (goose or duck), and pinot is classic with lamb. I don't really eat game, but I imagine that any of these would be good with game. The old rules about "white wine with light meat and red wine with red meat" is really not accurate--it was based on aesthetic appeal rather than compatibility. Any strong-flavored meat (such as beef or lamb), fish (such as salmon), or poultry (such as waterfowl) or heavily seasoned foods will be good with a red wine. Ditto with any robust vegetarian dishes. In fact, I have a recipe somewhere (if you are interested) for pinot planked salmon; basically, you soak the salmon in pinot noir for several hours, then grill it on an alder or cedar plank. Yum!
(can you tell my family is really into wines?
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Hope this helps!