Your mother may have been one of the few mothers to tell you not to talk to strangers as a child. I'm going to tell you the opposite. If you want to get a full travel experience, you need to talk to strangers.
Talking to strangers teaches me about local culture and history, and gives an excellent sense of what's really happening wherever I am. It's the reason I'm never lonely while traveling solo. It also regularly gets me free food, drinks, transport, places to stay, tickets to cool stuff, and invitations to the kinds of things you only hear rumors about in guide books. All I have to do is talk to someone I don't know yet. If you try it, you'll reap benefits, too.
A disclaimer: This does not mean you should wander up at night to the group of shifty-looking characters with baseball bats in a back alley and ask if they can break a $100 bill for you. Please be selective in who you talk to.
If the person makes you nervous, that's not always a bad sign. You just have to think about why you're nervous. If you're just nervous that the cute girl/guy at the bar won't like you, suck it up and go talk to them. If you think the old man on the porch won't speak your language or won't like people of your demographic, just be extra respectful, smile when you say hello, and judge further conversation based on his reaction. If you're nervous that bothering the guy wandering down the street at midnight swinging a machete might put you in physical danger, then maybe you should trust your instincts and go elsewhere.
I particularly encourage you to talk to local people. Fellow travelers are easy to talk to because you already have travel and being foreign in common. But locals are often more rewarding to meet. Ask for directions, instructions, and recommendations. It's flattering and you'll pick up information, maybe some new skills, and, if you click, a new friend or three.
If approaching random people on the street for that kind of thing scares you, we'll start somewhere easier. In fact, we'll start with four somewheres: your accommodation, on public transport, near tourist sites, and in nightlife areas.
For accommodation, it's going to be a lot easier if you stay somewhere with shared facilities than if you stay in a hotel. If you have a single room in the hotel, your opportunities are limited to the busy staff and people you see in hallways, elevators, and other places where extended conversation gets awkward, fast.
If you stay at a hostel on the other hand, you expand meeting places to a shared kitchen, common lounge most hostels come with, and of course the dorm you sleep in. Here are the magic words: “Hey, where are you from?” You can turn to anyone in any hostel anywhere and start a conversation, completely out of the blue, with those five English words. Even better, the staff are usually locals who like travelers, know the area, and often are more than happy to hang out and even show you around town after their shift is over if you take the time to actually talk to them.
As for transportation, especially on long train and boat rides, conversations spring up naturally if you're open to them. Everyone is going to be kind of bored and will be happy to talk to someone from out of town. Even if the “talk” is mostly gestures or passing a phrasebook back and forth. This is where you'll most often score free stuff like food or drinks. Just remember to share some of yours too.
Tourist sites, weirdly, are better places to meet people than you might think. Obviously you can meet tourists. But you should also talk to the staff, especially tour guides. They're mostly local people, most of them will speak English (and a few other languages to boot), and a lot of are often otherwise really bored and happy to have someone to chat with. You'd be surprised how little interest tourists seem to show in these people's lives outside of their jobs. Don't make that same mistake.
Finally, there's meeting people the same way a lot of people meet each other at home: nightlife districts. Pubs, bars, and clubs everywhere are places where you can, by unwritten law, strike a conversation with just about anyone. The only problem is that it's probably the most intimidating place to do it. If you're feeling self-conscious, just remember that 95% of the people you talk to are going to be worrying too much about what you think of them to pass any kind of judgment on you.
These are just a few places to get started. Don't let them limit you. You can talk to strangers just about anywhere you can find strangers, from in a public library to knocking on someone's door to ask to borrow some cooking ingredients. Unless stated otherwise by cultural taboo (see your travel guide or guidebook for details), they're all fair game.
If you're stuck for conversation starters, use props. One of my favorites is food. I've been a lot of places, and I have yet to find someone who doesn't smile when they're offered a cookie. Even if they turn it down, they'll often try to talk to you or offer you something of theirs within a few minutes. Another good prop is anything technological. If you've ever walked down the street with a friend who owns an iPhone or iPod Touch, you know how this works. My little netbook still gets me a lot of attention. But really anything interesting enough to elicit comment (though not offensively so), can work. I still remember walking down the street in Chicago with a bouquet of lilies and having every third woman I passed say something to me (mostly "ooh are they for ME?").
One last piece of advice. Think about the age of people you talk to. I was born in 1986. Almost anyone in the world my age or younger will speak some English, or be with someone else who does. However, if you've got the language skills, try to talk to older people. The older, the better. These are the people who lived the history of their homes and who can teach you more about the culture of a place than any of us youngsters can.
Now get out there and talk to people!