Contrary to what you've been told, Puerta del Sol is just about "THE GREATEST AREA" in Madrid. (Plaza Mayor and Huertas are great too) It has some of the best/coolest bars, restaurants, and nightclubs in the city.
It and the Plaza Mayor are THE tourist centers of the city and are not at all far from one another. Puerta del Sol does probably have a "claim to shame" for having the greatest number of pickpockets and purse snatchings but as far as violent crimes go they're almost non-existent, mainly because there are SO MANY PEOPLE walking around there day and night - not just tourists either. Most are locals.
The entire old center of town has problems with pickpockets and purse-snatchings but the great majority of visitors to Madrid never experience any of this. You can read a number of threads on this topic here in the Safety & Security fora.
But you really don't want to be too far away from this action-packed center because... well, if you stay further out and then venture in to enjoy the things worth seeing you'll still have to pass through on your way back. There are so many people in the streets day and night that chances are excellent that nothing will happen to you, particularly if you're in a couple or group. Just down walk the streets with your purse open or a visible wallet in a back pocket and you'll most likely be fine.
You might want to seek out some of the reactions/opinions from some of your countrymen (Canada) about Madrid and Spain. It seems that many don't care for it much. I can only guess it's a matter of perspective coming from a very clean and safe country like Canada.
Enjoy Madrid! We know you will! Saludos, MadridMan
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