This is a topic that in the boring hours at school I have spent an inordinate amount of time thinking about. Not that I'd ever actually do it (being a natural coward who runs from angry squirrels), but it is certainly food for thought: How to be a masked vigilante in the real world, and minimize your chances of getting killed doing it.
Training & Conditioning:
First of all, you would have to be in absolutely phenomenal physical condition. A vigilante has to be prepared for every possible situation, requiring a mix of strength, endurance, and speed equivilent to an Olympic decathlete. As such, it would be a good idea to roughly follow a decathlete's training regimen. There are probably people here who know a lot more about physical conditioning than me, so I will turn the floor over to them for this part to give pointers.
Second, you will also need to be an expert combatant, equal to or better than a military special forces operative. Gun skills are probably more important than hand to hand skills, but since you need to be prepared for everything, hand to hand should certainly be well trained. A combination of BJJ and Muay Thai is probably best, with an emphasis on the Thai kickboxing because your hands will likely be occupied with a firearm.
Finally, a high degree of education is a must, with an emphasis on law and criminal psychology. You want to be well aware of the laws regarding citizens' arrest in your area, and the psychology will help you to throw your opponents off guard. Batman actually has it right in that good scare tactics can take some of the edge off an enemy's skill.
An obvious career path then is through military or law enforcement. These programs will give you the skillsets you need to cover all your bases and at least partially get you on the track to the physical development you need. The problem comes in when you realize that the money required for the other parts does not dovetail well into those professions.
Equipment & Arsenal:
In the comics you can get away with using bolas, nets, boomerangs, and other archaic weapons. Not so much in the real world. There is a reason why guns have come to dominate modern weaponry. However, unless you want to be labeled a serial killer, guns carry their own problems. Therefore, less-than-lethal ammunition is best.
Again, since planning for a variety of situations is the goal, you want a variety of less-than-lethal ammunition. A 12-gauge shotgun is therefore the weapon of choice for the modern vigilante. The variety of stuff you can pack into a 12-gauge shell aside from buckshot is staggering. Aside from beanbag rounds, there are rubber rockets, hornet's nest, and the low-tech alternative: rock salt. Then there are utility munitions like dragon's breath, doorbusters, and flare rounds. The shotgun should be pump-action, since the sound of the pump produces a high intimidation factor (one reason why it is favored by police).
Another weapon favored by police are tonfas. In a fight, tonfas are designed for blocking and parrying, as well as delivering devastating blunt-force strikes. For this reason, they make for a good choice in a straight-up fight. Fighting fair isn't in your best interest, though, so it is a good idea to stack the deck in your favor via a taser and pepper spray as well. And when all else fails, steel-toed shoes and weighted gloves finish off your melee arsenal (as well as keeping your fingerprints clean).
To finish off the combat arsenal, it would be a good idea to pick up several utility weapons favored by police forces. Flash-bang grenades make for a great entrance. Tear gas and smoke grenades can further stack the odds in your favor. If things go awry, caltrops can help cover your escape or be used to put a quick end to a getaway car's tires. Finally, once you get someone in a grapple there is no quicker and more efficient way to incapacitate him than zip-tie handcuffs.
Now to the really fun part. No capes and cowls here, capes are too easily grabbed in a fight, and a mask can be pulled down over your eyes. A full helmet is best to protect the face and head, probably made of bulletproof plastic. Visor should incorporate night vision, and the whole setup should be air-filtered so you have no trouble with the tear gas. This sort of integrated gas mask would also help disguise your voice. Last but not least is a police band scanner built into your helmet, so you can easily tell when the law is on its way.
A cape is a good idea since it does help disguise the body outline, but its risks outweigh the benefits. Instead, a trenchcoat could serve the same purpose, plus has enough pockets to carry all the ammunition and equipment you need. As a last resort it should be modified for quick escape in case someone grabs you.
Underneath the coat, it almost goes without saying that you will need bulletproof armor. It's a careful balancing act between maneuverability and protection. Class IV plates provide the best protection, but they tend to be rather heavy and can degrade after multiple hits. Class III is probably a good compromise. You'd have to be sure to get leg and arm plates as well, otherwise one stray shot could incapacitate you.
Overall color scheme should be either dark red or blue. One could debate the benefits of one vs. the other for hours, but in the end it comes down to whether you would rather be a less open target (blue) or more intimidating (red). Either way, both are much better than black for nightime stealth. Beyond that, you would need to do a lot of experimenting to get a look that works.
Last but not least, transportation. At some point you will want to get away from the scene very quickly. The modern vigilante must value agility and speed in a getaway, not power, so an offroad-capable motorcycle is your best bet. Not only is it quick, but it is also easily stashed out of view, and can take shortcuts through alleys that larger vehicles can't.
Now, above all else, this requires you to be completely and utterly insane to even try. This is a horribly bad idea that could go wrong in so many ways. Still, it's fun to fantasize about.
Feel free to point out flaws in my reasoning or add some tips of your own.