If you ever lived in a busy city, you probably know all this already. If you use common sense, stay away from conflict zones and don't do anything stupid, you should be fine.
Take the advice from locals with a grain of salt, no matter how well intended. Most people travel very little and have misconceptions about safety, even in their own country. You may be told that a place is dangerous, you need a guide, etc., when actually it is very safe (they don't like the ethnic group living there). They may recommend a touristy place which is pleasant for them, but as a foreigner you will be targeted by scammers and aggressive sellers. Obviously, this is even more true for advice from non-locals.
Road crashes are by far the biggest danger to travelers: ~1.3 million deaths per year plus 20-50 million injuries. Avoid motorcycles and cars (see here, here, here). See here and here for country stats.
Your behaviour and attitude are important as they can give you away as a dumb, naive tourist. Dress and act like the locals.
Avoid expensive-looking clothes, backpack, gadgets, etc. A big backpack immediately draws attention. A cheap, small, normal-looking backpack is much better.
Don't look lost and confused. Walk confidently and act like someone living in the area minding your own business.
Traveling by private car in an unsafe region makes you a target for bandits and terrorists. It is much more discreet to use public transport.
TOUTS, SCAMMERS, ETC
These people can be a very annoying but don't let them spoil your opinion about the locals. In any country they are a tiny fraction of the population, but as a tourist they will always target you. Make sure you go outside the touristy places and hang out with ordinary people, not just with the touts. You will likely find them very kind and hospitable.
Touts ("hello friend!", "where you going?", etc): Don't respond, just keep walking.
Con artists, fake guides: Don't get into conversations with them. The longer you spend talking the harder it is to get rid of them later. If you think that your new "friend" is a scammer, make up some excuse and leave. A typical scammer is well dressed, speaks good English, knows a lot about your country, and is suspiciously friendly.
Sometimes it is easy to mistake genuine friendliness for a scam/hassle (and vice versa). Don't automatically be impolite to everyone who approaches you but try to get a feeling about the locals before engaging with them.
In most countries it is inappropriate to approach strangers on the street, especially of different gender. People who do so break the norms of their own society. You have no obligation to respond. Smile will be seen as an invitation.
Don't be polite to anyone who refuses to leave you alone.
Don't let a scammer intimidate you - their job is to manipulate you into willingly giving them your money, not to rob you by force. As long as they don't use violence, the police won't give them trouble.
If you are looking to buy something, avoid the sellers who specifically target tourists.
The reason why these people are so persistent is because most tourists eventually give them money, which only motivates them to bother the next tourist even more. Don't feel guilty about refusing, the people who harass tourists earn a lot more than many hard-working, honest people. If you want to be generous, donate to a charity or to poor people who don't harass anyone.
PICKPOCKETS
Don't be an easy target. Travel light and walk quickly. Be cautious when taking public transit and in large crowds where you have to slow down. Carry your bag on your chest in crowded places.
Avoid pants with slanted pockets that are easy to pick. Deep, jeans-style pockets are excellent.
A bag with a rusty zipper that always gets jammed is better than a smooth, silent one.
Pickpockets often have an accomplice whose job is to distract you. It might be someone bumping into you or spilling a drink on you, an old lady, a beggar blocking your way. Be alert when something unusual happens around you. Don't listen to any apologies, just keep walking.
Some less subtle pickpockets may grab your arms or surround you in a group. Keep moving, push your way through them or make some noise. If there are people nearby, most likely you will be left alone.
VIOLENCE
Violent crimes against travelers are very rare in most parts of the world.
If you feel you are being followed: walk into a shop, a restaurant, etc.; cross the street, head to lively area. Avoid dark alleys.
Don't fight, the attacker has more experience than you. Run away or yell for help in local language. If you can't escape, try to act calm and assertive. A thief has no interest in physically hurting you.
If you throw some money or a wallet at your attacker and run, he might not chase you. Consider carrying a wallet filled with small bills that you can give away.
If the attacker carries any weapon, don't resist. Avoid eye contact and hand over your belongings.
Carrying some kind of weapon with you is an easy way to get killed.