Here's my take:

First, there isn't much you as an individual can likely do against a well-prepared, well-trained terrorist, no matter how vigilant you are. Secondly, you have to decide for yourself how much you're willing to be ruled by fear (i.e. how much you're willing to be "terrorized" and thus let the terrorists win) by foregoing activities that you would normally enjoy (e.g. skip that Christmas market trip), or by treating people with suspicion that you otherwise wouldn't. This is a subjective, personal decision that I don't think anyone on Travel.SE can help you with. Having said that, I think vigilance normally means:

• Being very aware of suspicious items (abandoned bags/packages and the like), never touching them and ideally reporting them. Same goes for "suspicious" parked/abandoned cars (but of course it's hard to say what "suspicious" really means).
• Being aware of escape routes (locations of emergency exits and such) at all times wherever you are. If possible, pick locations near such exits.
• Being aware of individuals behaving suspiciously / differently from the surrounding crowd (this is difficult, but has added benefits of spotting pickpockets and the like)
• If at all possible, avoiding crowded events (or at least the most crowded places at crowded events)

Learning basic first aid can be helpful too.

I have been continuing to research this and here is what I have determined:

Before you go

In the past I would cheerfully go to Europe with my cell phone working only as a wifi device. This isn't possible if you need to exercise tremendous vigilance and caution. Adjust your travel budget to include getting a SIM card for your phone. Research in advance where to get one - ideally the airport on arrival. Learn the emergency number for the country where you are going - it isn't always 911, though some local SIMs will dial that for you if you dial 911, don't count on that.

Register with your government (example for Canadians) so that they can contact you in case of trouble.

Find that one friend or family member we all have who is always online, on Facebook, on Twitter, has news apps, turns on the all-news channel when they fold laundry, and tell them where you're going. Not just "Europe" or "Asia", not even just what country, but what city. Bristol. Brussels. Shanghai. Melbourne. Let them know that if they hear anything specific to your city (bomb went off in the northern suburb) that you would be grateful if they passed it along, day or night.

Set up automated alerts (with Google News or various news apps) for your destination, your airline, your hotel, and so on.

Check your country's travel warnings (examples for Canadians and Americans) and consider adjusting your plans. Get the contact information for your embassy or consulate in your destination.

In addition to the usual language learning (hello, thankyou, please) for your destination, learn "help", "danger", and "police". Learn how to spell Police and other "first responder" words that may be on vehicles.

On Arrival

Get that SIM card as planned. Update your government, your "newsy" contact, and at least one close friend or family member and give them the number. Put your contacts into the phone if they aren't carried over. Include the embassy and consulate information, and also your hotel - both so people who are helping you can find it, and because the hotel staff may be able to help you over the phone if bad things happen.

Observe what normal is for the country you are visiting. In the South Pacific it is perfectly normal to walk along the road with an unsheathed machete. In downtown New York, not so much. Observe how other people react to what you are seeing and set a baseline. For me, seeing long guns is a bit of a shock even when they are in the hands of uniformed officers. Also observe what police and soldiers wear. You are likely to see some at the airport, so don't waste that chance.

Establish a source of local news - whether that's a website, a tv channel, or the radio - and get in the habit of checking it at least morning and evening.

Daily

Be prepared to adjust your plans if a destination (or the transportation you would have used to go there) is closed. Monitor your news source, and reassure your home contacts you are fine (a cheerful tweet that doesn't say "I am not dead yet" but instead raves about the seafood or the architecture, a facebook status that shows you smiling at a landmark, etc.) You don't want to worry them by explicitly mentioning that you feel the need to report your nondeath, but you do want to show them that you're fine.

If you are traveling with someone else, and especially if you don't each have your own phone, put a piece of paper in each person's pocket each day that has your name, where you are from, where you are staying, and the local cell phone number, along with the names of your companions. Ideally in English and the local language/script. Even a conscious person will appreciate just being able to give this to a first responder or helper rather than struggling in the local language, and for an unconscious person it could be vital. Some people make sure to take pictures of each person on all the cameras and phones each morning, so you can remember what each other are wearing, or show the picture to helpers. It can't hurt much to do this, and you'll be sure to have a few pictures of yourselves in this lovely destination.

When Out and About

Be fully present at your destination. Don't read your phone while you travel on the subway or wait in a line, for example. (But don't set it to silent non-vibrate - you may get an alert that you need to read.) Observe those around you. Talk to those you are with, but from time to time, pause to see what else is happening.

Don't just randomly wander. Some days, getting on the subway and getting off at an interesting looking station is great fun. Or seeing what is north of the museum. If you want to do that, at least pay attention to what you are doing. Have a map with you and know where you are on it. Don't assume you can bop about for hours, have no clue where the hotel is, and use a cab to get back at the end of the day. If things go very bad, being lost and hoping a cab can rescue you isn't a good strategy. So just check the map from time to time on your adventure and think about how you would get to the hotel from where you are now, that sort of thing.

Try not to rein yourself in too much. Many people warn that too much caution means "the terrorists win" and they may have a point. If you won't eat on a restaurant patio or terrace because people were shot at a handful of Paris terraces, and especially if you won't do so on a Friday night, then you are doing precisely what "they" want. A coward dies a thousand deaths, said Kipling, and the brave man dies but once. Walking in constant fear won't keep you alive, it will just make you unhappy.

If you see something

It's easy to tell you "say something" but in a foreign land that is not simple. If what you see is unambiguously awful - someone is shooting into the crowd - focus on running or hiding, but do call the emergency number as soon as you can. If it's just something iffy, the fastest thing to do is grab a passerby and point it out to them. An abandoned bag, three guys standing on a roof staring at the crowd, that sort of thing. If there is no passerby, scan for a police officer or soldier who you can inform. And try to get yourself out of the area at the same time. Call the emergency number on the way.

If it all goes very bad

We are now out of the realm of vigilance. Your top focus should be protecting yourself and keeping track of those you are with. Afterwards, you will want to reconnect with those you are with, if you were separated, and inform your home people what has happened and how you are. The paper in your pocket and the SIM in your phone should make this much quicker and simpler than it otherwise would be. Be aware that Twitter is often of tremendous practical use during a crisis - consider the #PorteOuverte hashtag, for example.

It all depends on where you're going and how risk-averse you are.

Basic precautions

• Don’t keep your passport, credit cards and other valuables in the same place; use the inside compartments in bags where possible. Carry your bag across your body rather than on your shoulder. Keep digitized copies somewhere safe.

• Situational awareness, being aware of your surroundings. It's very easy to end up on a empty street, or in a less than recommendable neighborhood, to be separated from your group. Keep an eye on your stuff, who and what kind of people are around you, the point here is to be able to react fast enough should anything happen.

• Avoid substances that decrease alertness, like alcohol and drugs, you're less in control and less aware of your environment, your reaction time is delayed and your judgment is impaired.

• Avoid being trapped in crowded places like a subway at rush hour or places of public gathering, demonstrations, political gatherings, things can get unpredictable in the event of something serious (or not !) and your movement is limited.

• The usual advice advertised everywhere on subways and trains is, if you see something suspicious, report it. Obviously this depends on your understanding of suspicion, a bag without its owner on the tracks, shifty looking individuals, an unusual event you think is meant as a distraction or trap ...

Paranoid precautions

First of all, you should realize that as a Westerner or a foreigner you're more likely to be targeted, although most crime and attacks tend to be indiscriminate.

• Keep local emergency numbers on you
• Keep your ICE (in case of emergency) contact list up to date
• Know the location of your country's foreign mission in that country, you could possibly seek refuge there should things go bad
• If you don't know the local language/customs, don't leave without a local friend
• Avoid staying out very late, returning alone in taxis
• Know regular and emergency exits in any place
• Knowing basic first aid can be life saving
• You can have someone regularly text you or check your location via GPS, I know some foreign correspondents or extreme explorers are required to check in via GPS every 24 hours and notify someone of their whereabouts at all times.
• If you notice that a security operation is underway you should immediately leave the area, bystanders also get killed.
• Monitor local media and follow the advice of local authorities.