Become a fan of h2g2
Now that you know what you're looking for, you need to now how to go about looking for them. Once again some people may say you need a really big, expensive telescope and different filters to properly star gaze. And once again, I have to say they're even more clueless than my pet rock. You don't need to spend a lot af money to star gaze, actually, you don't need to spend any money at all. All you need to properly star gaze is a clear night sky, something to lay or sit on, and glasses(if you were already in need of them to get around in the day). Sure telescopes are nice and all, and if you have the money and really want to buy one, then have at it. I won't stand in you're way.
Now that you know what you're looking for and how to look for it, you may run across a few common problems that we all run into every now and then.
One of the biggest problems people have when star gazing is that they live in the city and it's very hard to see any stars, no matter how late it is. The best solution I have found for this is to leave the city. Take a weekend off with you're freinds, family, significant other, or just by your self. Go out to the country and stay in a motel, bed and breakfast, with freinds or family, or if you really want an adventure, try camping.
Another problem I've encountered personally is finding something to sit/lay on. A blanket works well for most situations, but if it has recently rained, or the ground is just wet for some unknown reason, then I suggust not using a blanket unless you really like mud. You could use lawn chairs or other lightweight, moveable seating if you don't want to lie down, however if you have to lie down in order to really enjoy star gazeing and the ground is indeed wet, then you might look in to buying a nice inflatable mattress. Inflatable mattresses are fairly inexpensive, but aren't always that convenient since you have to inflate them your self. Or if you would prefer something a little more sturdy, then a sun lounger may be the perfect solution. They're fairly inexpensive and pretty easy to find in the summer. They're a lot easier to drag aroung too, just fold up and go.
Stargazing can be enjoyable for many people, but it is understandable that even if you follow these directions you may not have as wonderful night as you had hoped. If this occurs, pick another night and try again. Maybe things will be better, but if not, don't feel bad. Not every one enjoys stargazing. At least you tried it out, and you know you're not missing out on any thing would really like. Good luck, and remember to dress warmly; the night air can get pretty cool.