I third the rec for Mirassou Pinot Noir.

Cline Pinot Gris (smooth and fruity apple/pear noats)
Fire Road Sauvignon Blanc (crisp, dry, passion fruit and green fruits)
Also second the menage trois red

Anything from Cline, in Sonoma, is good. Their basics start around $10, and some go up to $20.

For just under $10, Blackstone reds are yummy.

Good starter reds for $12-$15 are Riojas. (It's a Spanish appellation--look for the word "Crianza." "Reservas" will be more.)
Marques de Caceres and LAN make good ones that are widely available.

Our very fave, Pine Ridge Chenin Blanc/Viognier blend, is not quite as cheap as it used to be. You used to be able to find it anywhere for $11-12, now you have to look for it on sale for $14-15. Still a steal. Goes with everything.

PP's suggestions of Hogue and Chateau St. Michelle Riesling are also good.

If you live in Virginia, go to Total Wine and ask anyone there for recommendations. You'll find better stuff than many of the mass-produced recs here, and can choose your price range. Lots of tasty stuff for under $10.
Costco's under-$15 stuff tends to be mass-produced, but if you spend more than $15 they have some nice stuff. Not so much in-person guidance, though.

Also, check out Wines Till Sold Out: www.wtso.com.

Hogue Chardonnay
Mezza Corona Pinot Grigio

Mendoza (Argentina) Malbecs: Gascon, Kaiken, Dona Paula. All under 12.00 depending on where you get them. All lovely wines.

Lately, Oyster Bay Sauvignon Blanc - but really any NZ Sauvignon Blanc works for me.

Chateau St. Michelle Riesling (they are the world's largest producer of Riesling - crazy!), and their unoaked Chardonnay is really good, too.

World Market is a great place for wines, by the way, I haven't gotten a bad wine there yet!

Pacific Northwest:
(Washington) Chateau Ste Michelle and Columbia Crest make wonderful Riesling, Gewurztraminer, and Pinot Gris

California:
Not quite as inexpensive at the others, but still under $20/bottle
Ravenswood reds in particular the Zinfandel and Syrah

Australia:
Many Australian wines are pretty inexpensive and quite good. Here are some of my favorites:
Penfold's (does good reds)
Wolf Blass (their yellow label Cabernet Sauvignon is possibly the best discount red wine I've had)
Wyndham Estates
Lindeman's does good inexpensive whites

Kim Crawford Sauv Blanc
Rodney Strong Cab Sauv
Alamos Malbec
Pacific Rim Riesling
Mirassou Pinot Noir

I go see Steve Long at Out of Site Wines in Vienna. He hooks me up with lots of both reds and whites that are much better than these mass distributed ones, and are in the $10 to $15 range.

I like Chilean and Australian shiraz and malbec, and Cameron Hughes whites and reds.

Apothic Red= about $14

and though not as inexpensive at $22 per bottle (I bet you can get it for under 20 if you shop around): Seven Deadly Zins

Red Diamond Cabernet & Columbia Crest wines are great. Total Wine has the best prices on them. Usually under $10 per bottle.

But for real any new zealand savignon blanc under 9 is excellent.