Arguably the best (and definitely the most affordable) way to get a taste for the 'real' Cancun is by flagging down a public bus anywhere between the main districts of Zona Hotelera and El Centro.
As an added bonus, if you're travelling after dark the buses turn into something of a mobile theatre, with jugglers, musicians, clowns and a diversity of street performers looking to earn a few pesos.
Tours of Cancun itself are almost non-existent, so catching one of the 24-hour buses (which generally show up every five minutes or so) is the easiest way to see the city sights.
If you're keen on exploring downtown El Centro on foot, the experience can be a little disorienting in the most enjoyable way. Lose yourself in the circular districts known as Super Manzanas and discover the hidden gems that await!