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.
The intermittent reinforcement factor at work which lends to 'found one' and makes it interesting (I am guessing..)
Wine is very similar. Price does not equate quality. That said, a very good $10 bottle of wine will be less quality than a very good $30 bottle. It's confused by the fact that there are some pretty outstanding $10 bottles and some pretty horrible $30 bottles.
I used to have a $10 favorite that is really solid but started venturing into $15-20 territory and there is a definite quality difference you might not notice if you are new to wine.
But wine is far more pretentious than beer and it's best not to get wrapped up in that world. Just drink what you feel tastes good. You'll notice that as you drink more, your tastes change and it's good to try new things.
It's pointless to drink a bunch of expensive wine if you haven't experienced much. It takes time to develop an understanding of the varieties and flavors to look for in order to really appreciate what a more expensive wine may offer (or an outstanding cheaper wine!)