I've been to Mexico City, Tampico, Cancun, Tulum, Puerto Vallarta and Manzanillo. I've had very few problems and the one I did have was while driving during the day; we were speeding a little over the limit.
The main dangers are not to drive at night. There are unscrupulous people out there who prey on unwary tourists. Don't speed, ever. If you're caught, just pay the bribe or 'mordida'. As well, the roads are bad and in some places full of huge pot holes. And sometimes animals cross the road at night.
Next, do not get involved with any Mexican person that you feel might not be trustworthy and don't buy drugs. It's possible to be caught in drug gang problems. Sometimes, one can be in the wrong place at the wrong time but that can happen anywhere. I think, even though Cancun is known as a major drug route, it is very safe because everyone wants to protect the booming tourist trade there. Puerto Vallarta is overrun with time share salespeople and beach vendors so mostly just irritating.
Otherwise, we saw signs for crocodiles along the Blvd. Kulkulcan in Cancun but never saw any actual crocs. If you have a sensitive stomach, be careful of street food but I eat tacos and tamales whenever I go and am never sick. If you're worried, take out travel insurance before you go. I may be lucky in that Mexico really agrees with me!
I wrote a hub about safe travel in Manzanillo, a sleepy town south of Puerto Vallarta, which I consider to be very safe as the Mexican navy is headquartered there and the state of Colima has very low crime. If you want to try a first visit, Manzanillo might be a good bet.