I like Meridian’s Merlot.
Go for the classic: Gallo Hearty Burgundy. Cheap and surprisingly good, and it goes with all sorts of things.
Honestly, avoid the mass produced ones. Texas has some nice wineries. Find one that makes a good red with concord grapes. They generally make a good ‘juicy’ wine with a small alcohol taste.
I’d say try a Spanish wine if you want a good, cheap steak wine. The Spanish Rioja wine is loosely similar to an Italian Chianti. (Both Rioja and Chianti refer to a region that produces a particular type of wine, not a particular maker.) Riojas are typically big and spicy. Very good with a steak. When looking for this wine, obviously look for “Imported from Spain”, and look for the word “Rioja” fairly prominently on the label, and from your OP it probably doesn’t matter to you who the maker is, so just pick one (but Marques de Caceres are consistently good). Also, the best part about these wines is that even though they are imported, they are really cheap because alot of people don’t know about them. They are about half as expensive as a comparable Italian wine. You can get a solid tasting bottle for about $10, and a really quality bottle for around $20.
Thanks for the advice, folks - I took notes. 
We’ll check some of these out. Ought to get our first bottle of wine this week; we’re having dinner in on Thursday or Saturday. (Long delay because I work third shift and so rarely have dinner home.)
I’d go with Kendall Jackson or Sutter Home if you want quality/low cost. Check out your local news website’s dining section for some tips - I hear that French wines are doing quite now, and you can’t go wrong with Aussie wines.
Cheers!
Cheap cheap wine: Boone’s Farm Strawberry Hill or Sangria.
Good tasting cheap wine for a steak dinner: Riunite Lambrusco – red, dry, and has a great aftertaste.
Another vote for Yellowtail Shiraz - I’m not a drinker at all, but someone recommended it to me to serve to friends who DO drink wine, and they all loved it.
And another friend always brings me a bottle of Luna Di Luna Merlot-Cabernet because the pretty bottle matches my red kitchen I get to keep the bottle, and everyone else seems to enjoy the wine.
Red – Shiraz.
White – White Zinfandel.
So far in my experience, it’s hard to go wrong with either one, regardless of brand (not to say that some brands aren’t better than others, of course). I would expect prices to be in the $5-10 range.
We’ve decided on the Yellowtail Shiraz as our first attempt. Is this something that’s readily available, or will we be going to a liquor store? (Or rather, will Gun be going to a liquor store while I’m at work?)
My local grocery carries it, and it’s my understanding that it’s pretty popular right now, so it’s likely to turn up wherever you’d buy any other wine.
Ever try Japanese plum wine? Great stuff, doesn’t taste alcoholic at all. It smells lovely too. Just stay away the from the Kikkoman brand.
I second the recommendation of Riesling, although I can’t remember the name of the brand I prefer off the top of my head.
If you were in Cali, I’d just say go to your local grocery to find Yellowtail, it’s a pretty common brand, but I don’t know how it is in Texas. If you have a Cost Plus World Market around, they should have it. I don’t recommend Yellowtail’s Chardonney, it’s not bad, just thought it was kind of blah.
It was, I just can’t remember the name of it. I got a bottle of the blackberry merlot trying to re-find the good one, and I don’t think that was it.
Good call on the Yellowtail, I’m a big fan of that label, and I hear they’re really taking the market by storm, so you should be able to find it. When I mentioned to my brother (who’s in the industry) that I’d tried this great new Australian brand, the first word out of his mouth was ‘Yellowtail.’
I drink wine about once a week, maybe, and I pretty much never go over $10 a bottle, so there are good values out there. You will get a stinker from time to time, sometimes it’s just a bad brand, sometimes a bad cork will do it too. I’d stay away from the big bottles, wine doesn’t keep too well.