Assuming you're in the US and have a favorable tax environment, you might have some luck with Montepulciano d'Abruzzo. It's generally an earthy wine with relatively low acidity and tannins which means it will be easy going. It's usually more interesting than similarly priced New World wines. It seems pretty hardy as well -- I've only bought one bottle that was faulted (out of many, many bottles) and that was just a little Brett and it was still drinkable.

You might also be able to find a $10 Chianti Classico. That will give you more structure which will make the wine feel a bit more "refined".

Also check out Cotes du Rhone reds. These will often be cheaper than the more famous regions in France but share a lot of the "refined" characteristics of their more well known counterparts. One of the best wines I've had recently was Cotes du Rhone (sorry, didn't get a pic of the label) and held up favorably to a Burgundy I had brought (to the lunch) at 4x the price.

Echoing what someone else wrote here, find a local wine seller. Wine is such a complex topic you really need someone to point you in the right direction. Wine is also a great way to meet interesting people. A local wine seller is a good place to start.