So you want to become an oenophile (a lover or connaisseur of wine). You don't have to be a wine-maker or have a basement cellar in order to appreciate fine wine. Just follow these tips to start you on the road to a new hobby.

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Get information from the experts. Read books on wine. Purchase wine guides. Subscribe to wine magazines.
Go to a wine shop and ask the staff for recommendations. Look for bottles of wine with write-ups near them, award citations and high magazine ratings. Try to go when you know the store is holding a tasting with samples.
Attend a local wine tasting or a wine appreciation class. These are held at adult schools, winemaking schools and fine restaurants.
Join a wine group.
Visit a winery. You'll learn how wine is made, see how the grapes are grown and be taught the proper procedure for drinking wine.
Buy wines that match the taste of the food you're serving.
Don't just stick to traditional reds and whites. Try sparkling wines, ice wines and dessert wines as well. Try wines that aren't just from Italy, France and the Napa Valley. Try wines from New Zealand. For American wines, try South Dakota or Idaho. Internationally, try wines from Argentina, Portugal and Australia.
Learn about different grape varieties. Traditionally fine wine was made from mainly French grape varieties, but now a much wider range of grape varieties are being used.
Subscribe to online wine newsletters. They are free and informative.
Look for a wine school in your area. Most host courses or tastings. Local adult schools and restaurants also hold wine appreciation classes.
Have an informal tasting either at a friends house or a BYOB restaurant where each person brings a different bottle of wine. This way you can taste a bunch of different things without spending a lot of money. And you get a great deal of wine.
Make wine inexpensively at home. There are starter kits at homebrew supply stores or online; you learn about gravity, yeast, fermentation stages, clarification, and adjust yeast and flavorings such as oak. Wine's taste changes most quickly in the first few months of fermentation.

[edit] Tips

You don't have to spend a ton of money to find a good bottle of wine.
Try making your own wine from grapes if you live in an area with access to wine grapes. If you don't, fruits or even grape concentrate kits are a good alternative. Basic small scale amateur winemaking equipment and supplies are easily available. Making your own wine will help you understand much more about wine, plus you will have your own "estate bottled" wine to drink!
Keep a wine notebook to bring to tastings so you can write down what you like.