What I want to do is taste red wine varietals. Merlot, cabernet sauvignon, zinfandel, syrah, and one more which I have not decided upon
If you want to do around $20, don't overlook Australia and South America.
You can get merlot for around $20 in the US too - Chateau St Michelle is a good choice. They have several bottlings and they're widely available and show good typicity. Columbia Crest too, but see if you can find the reserve bottlings. They're around $20 - $25 and quite good for the price. Whitehall Lane also makes a good reserve merlot for around the same price.
All of the above make cabs too and you may want to look for those. Not to get into the "typical" character of the various grapes, but sometimes they are so plush and ripe in CA or some parts of Australia that you can't really tell what you have. Gross generalization, but WA will give you better quality for the money at your price range.
However, if you look around outside of Napa, CA has good cabs too. Beckmen if you can find it is a good intro. Beaulieau Vineyards makes several lines but the Rutherford cab is one of the best values in Napa and readily available. It's not crap wine at all and it's a classic American cab so I'd include it if I were you. Their Coastal Series or Napa series aren't much to talk about however, so make sure you get the Rutherford.
And Chile is making some cabs that people really like. I tend not to because they always seem to have a green note. But they are typical. Casa Lapostole Alexander Valley merlot is a perfect example of the grape and their cab is likewise. They are more "French" in style.
You can also look to Bordeaux, but unless you know the wine, you'll likely get a blend and you wanted varietal specific wines. However, you can get straight syrah from France and that may be something to consider. If you can find a Crozes-Hermitage, that would be excellent. Those are usually the lower end of North Rhone in terms of pricing, but with the Euro today, they may be over your budget. So you can look around for something from the Languedoc or South France. France is where syrah came from so it would be good to find one. However, Australia has made syrah it's signature grape and you can find hundreds at your price point. They will be different in style for the most part, so you need to decide which you like. Just for fun, I'd probably include one really ripe job from Barossa and one more restrained from the Rhone and see if people could detect any similarity. And don't overlook South Africa - they have some decent ones in your price range, although they still turn out a lot of swill at that range IMO.
Zin - there are too many to count and it's really only important in the US, so you need to look for something from CA.
For the last grape, you'll get all kinds of suggestions. Pinot Noir is a pretty good choice actually, since it's available, popular, and bottled as a monovarietal and rarely blended. It's probably a good choice because it's also made in many different countries. You can sometimes find a generic Santenay that drinks well. I'd put in something like that because it will give people an idea of the grape and it will still have some character from the area.
Anyhow, just my 2 cts. Good luck and have fun.