I really don't care for red wine, and I have been told that it's probably the tannins that I object to. If that's the case, what varieties of red wine should I look for when a recipe (usually a stew or sauce) calls for "dry red wine"?
The difference between red wines and white wines are that red wines are fermented with some of the skins, while white wines are not. Tannins come from the skins, and make people's mouth feel dry, hence more tannins mean a wine is called dry. So you cannot have a low tannin dry wine, as dry means tannic by definition. Much of a wine's characteristics come from the properties of the earth, water, and climate that the grapes are grown in (the french call the sum of these characteristics terroir), so you can get a dry wine from most grapes depending these factors. However, some grapes produce less tannic skins, and therefore overall produce less tannic, ie fruitier wines.
Common fruitier red varieties are pinot noir, garancha (granache), primitivo (Zinfandel), and pinotage. All these varieties should be available in US supermarkets. If you can find an Oregon pinot noir then that would be my choice, they are very good value and great tasting wines. California zinfandel used to be considered cheap plonk (because it was), but lately has had a resurgence and there are some fantastic wines from this grape.