Brands or varietals?
I love old world wines from the classic wine producing regions - Bordeaux, Brouilly, Burgundy, Champagne, and Rhone Valley in France; Rioja in Spain; Moselle Valley in Germany; Tuscany in Italy. I'm mostly a red wine drinker and like bigger, bolder flavors typical of Bordeaux grapes - Cabernet Sauvignon, Cabernet Franc, Merlot, Malbec, and Petit Verdot. For whites I like more Italian dry wines (Pinot Grigio) or frutier French wines (Chablis).
New world stuff is fun to try too and there is some great stuff out there - Chile and Argentina do some great stuff with Italian grapes (Bonardo), and American wines keep getting better. Outside of California, Oregon, and Washington; Virginia (great soil for Cab Franc and Petit Verdot), New York, Ohio, Texas, Kentucky, and New Jersey are starting to take wine making seriously. I can't vouch for others listed, but I belong to two wineries in Virginia and have enjoyed learning about less well known, but more accessible local varietals (Seyval Blanc, Norton, etc.).
One of the best things about wine is learning, and tasting, new things, and understanding a bit about how it's made, and how soil, water, and sun can effect and change things. I would recommend taking a 'beginner's course' in wine tasting - not to be a snob or anything, but so you can appreciate some of the details of wine tasting - the nose, the finish, the acidity, sugar content, etc., and coming up with your own tasting notes.
To get started with your preferences, I'd say look for Pinot Noir, Brouilly, or Beaujolais. If you want a bit heavier, but still sweet, look for port style wines. They're typically sold as 'desert' wines.
