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Santiago is notoriously infamous for the smog and it's worse during the winter May-September. The locals welcome the rain which falls during winters as it cleans the air. Be sure to carry bottled water with you during the summer and avoid food or drink from street-vendors. Be prepared for sauna-heat on the metro during summer.
By South American standards Santiago is a safe city, but visitors should be aware of pickpocketing and other petty crimes. Avoid parks at night and don't wear expensive looking jewelry or watches even during the day, unless you are in Las Condes or Vitacura. If you're alone, avoid large crowds of people, especially downtown.
If you happen to have bad luck and get robbed, do as you're told by the criminal and if you don't understand Spanish, give away the wallet. Not doing so can provoke an attack until you give away your wallet. Don't try to stand up to them and once again: do as you're told.
Don't flash your camera, take a photo and hide it while not using it. If you're getting robbed and the criminal has spotted the camera expect to give it away too if you want to stay out of trouble.
If someone approaches you on the streets and promises to get you better chances at changing your dollars/euros into Chilean pesos, NEVER accept their deals. They're con-people who take advantage of foreigners not knowing the details about currency and confuse them with big words to take their cash away. Only change your money in legal currency exchange centers, which may take more time but are much more trustworthy. There's one in the Airport, but they are also easy to find in the downtown and financial areas, or in the malls.
If you are going to see a football match, be careful with the "barras bravas" who are the most fanatic but also dangerous fans. They are often involved in troubles with the police both inside the stadium and outside. Avoid buying tickets in the sections where the bravas dominate which it often does behind the goals. The middle section is the safest but if you have a friend who wants to support Colo-Colo and another Universidad de Chile for example, avoid showing it. Even if the middle section is safe, showing different shirts can cause you trouble. Either go with the same shirts or dress neutral. Other football matches not being the "Superclásico" between Universidad de Chile against Colo-Colo should be pretty safe.
Walking to the stadium you will find people begging for some pesos so they can see the match. Avoid giving them if you want to stay out of trouble.
The barrio where the Estadio Nacional is located is a place which is normally peaceful, but you have to walk with precaution and keep your eyes on other people when it's crowded. It's better to take a taxi to the arena, or a rental car if you can find a place to park it.
There are some neighbourhoods/barrios that should be avoided. Even if few carabineros and locals speak English, they are the ones who know which areas are to be avoided since some of them can actually be accessed by metro. Some people at local Starbucks are more likely to speak English.
If you see fewer tall buildings and more houses with locked windows and entrances, then turn back. The changes happen very slowly, unlike some other Latin American cities, so you'd have to travel quite a lot from a wealthy safe neighbourhood to a dangerous ghetto. Avoid especially La Legüa not to be confused with La Ligua in the V Región which is famed in Chile for its high crime rates and single police cars won't even enter the area.
Try staying away from the following comunas: Lo Espejo, La Pintana, Puente Alto especially the Plaza de Armas, La Cisterna, San JoaquÃn, El Bosque not to be confused with a neighbourhood located around metro El Golf, which is in Las Condes, San Ramón and La Granja unless you know exactly where you're going. Though these places are not completely unsafe for the most part, they can have a few unsafe spots and don't have much touristic significance.
The safest comunas are Providencia, Vitacura and Las Condes. All of them have lots of local security guards, besides Carabineros, and locals are more likely to speak english, especially young people. They are not completely safe, though: petty theft still takes place, so keep your eyes open in the streets. Lo Barnechea can be tricky as it is the only comuna that has both extremely wealthy and extremely poor neighbourhoods ever since Pinochet's dictatorship; "La Dehesa" is wealthy and safe, "Cerro Dieciocho" is as dangerous as La Legua.
Overall, Santiago is very safe if you travel by car.
The metro is regarded as safer for travelling amongst the locals, even though security has increased in the buses after the introduction of TranSantiago. But some locals still prefer using the metro especially when it gets darker, since almost all the stations have guards. Don't expect the staff to speak much English.
In any situation, you can trust in the Chilean Police Carabineros. Although you can hardly find one who can speak English, they will try to answer your questions, solve your problems or give you orientations. DO NOT ATTEMPT TO BRIBE Chilean Police. Also remember that the Chilean police is a militarized police. Therefore the police special forces can be violent or unreasonable, be careful.