Find a wine shop or bar that does tasting events. Typically a tasting will concentrate on a particular region or grape, and go through 6 to 9 small glasses. Sometimes they'll be done blind, but more often you'll be talked through what to look for.
Typically you'll also have some cheese and cold meats to eat - useful to see how the wine reacts to other flavours. For example, malolactic fermentation in a wine brings out a unctuous creamy taste from a good cheese, while other wines make meat more savoury. Pâté is another interesting pairing: different wines can bring out different flavours from the same substance.
Wine shops will typically be doing tastings from bottles that they can sell, so they won't be super expensive (but probably more than $10). But rank your own preferences, and notice how your preferences change as you taste different wines - it can happen that a wine that seemed tasty to begin with pales after you've had a different wine.
It's a personal journey, IMO, and it's a long enjoyable road.
