I work part time in a winery in the tasting and production rooms, so here are some of the guidelines I tell rookies who come in to our winery for the first time:
1) Drink the wine you like - sweet, dry, heavy red, whatever. There is no "good" or "bad" wine, there is only what you like. Buy what you like and don't apologize for it.
2) Only taste a few wines at each winery, as opposed to a little of everything, as it kills your palate and soon you are unable to tell the difference between wines, and the alcohol clouds your judgement as to what you taste in the wine.
3) Write your own tasting notes - include what the wine smells like. Cherries? Tobacco? Earth? What you smell is what you smell, but practice with your nose. Same for what it tastes like--don't read the tasting notes they give you, practice analyzing it for yourself.
4) Taste the sweet wines first, and move towards the stronger reds. Don't be afraid to pour some out, it's smart, not insulting (the difference between "tasting" and "drinking").
5) Ask questions! That's what we're here for in the tasting rooms. Most are quite knowledgeable, and if it's not busy, the smaller wineries can give you tours of the production room, maybe give you a taste of the barrels, etc. Have fun!
Depending on where you live, there are often multiple opportunities to educate yourself in the art and knowledge of enjoying wine.
Community College Classes: any community college with a culinary arts department likely offers a wine-tasting class. I took one for fun one year. It was $40 for 6 weeks, and we met every Thursday night for 2 hours. The teacher brought between 7 and 10 different varietals for us to taste and also provided fruit, chocolate and cheese. She would show us how different foods brought out different flavors in the wine. We were taught what to expect in the aroma and on the pallet. It was a lot of fun and though no one walked away an expert, we certainly had a better idea of what we liked or disliked at the end.
Free opportunities: Most of the following examples do not have any overhead or "entry" fee, however, most expect that you purchase a glass or a bottle of wine before you leave.
Upscale grocery stores: Many (such as Whole Foods and The Fresh Market) have weekly wine tastings at a scheduled time. Even if you do not live near one of these stores, check with any grocery store near you that has a wine section and ask if they do tastings. Usually these are free, and in my experience, not very busy. You can get lots of personal attention and have questions answered. Also, it is very low key, so do not be afraid to admit you know nothing about wine. Ususally this kind of customer is exactly who they are hoping for.
Wine warehouses and/or wine-bars: Usually in slightly bigger cities you can find stores completely devoted to wine (like Total Wine and More) as well as restaurants or wine-bars that specialize in wine and wine-pairing. These places are just as excited to educate their customers as they are to sell them something. In addition to holding their own scheduled wine-tastings, I'm fairly certain you can call ahead and schedule a "wine tasting party" yourself, especially if you are bringing in a group of paying customers.
Local vineyards: If you live in an area of the country where grape-vines grow and local wine is abundant, you can take a tour of local vineyards. Washington, California, Texas and North Carolina are four states that I know of where you can pick up a map of local vineyards at any tourism venue and visit the very place where the grapes grow.
Do it yourself: In the novelty or "Food" section of any major bookstore, you can find "Wine Journals." These are notebooks organized specifically to help people keep track of different wines they have tried. You enter the date and the occasion, the wine brand and varietal, and what you ate. Then there's a section to describe your thoughts. This is something that is fun if you want to go out on your own and just try. In the end, any expert will tell you that the definition of a "good wine" is simply one that you enjoy.