One of the quickest, easiest, simpliest and safest ways to travel around the Hotel Zone is via taxi. The drivers know all of the main attractions around Cancún and so you may even get an impromptu commentary on the journey. You will certainly have a captive local, who is usually happy to share tips and stories.
There are universally agreed taxi fares in Cancún. Many of the hotels display them, on a board outside their doors. It is worth reading these boards and having a mental check-list of what you should be paying. Some taxi drivers aren't above overcharging, if (s)he thinks that you don't know the going rate. Ask the taxi driver the price of the fare before you get in and, though not mandatory, remember to tip him/her afterwards, if you received a good service. If you're travelling within Cancún, then a reasonable tip is 10-20 pesos ($1 USD). If you're travelling to or from the airport or one of the attractions within the city, then tip a little more. Obviously the amount you choose will depend upon the quality of the service. Don't feel beholden to tip, if you feel that the service was bad.
Incidentally, a taxi driver usually won't give you any change, so try to pay the exact amount. They do actually carry change though, so if you wish to receive some, please do press the point.
Taxis may line up at ranks, which are usually signposted with a frontal view of a taxi, in white, against a blue background. Alternatively, they can be flagged down with an upraised arm in the street.
Note that the taxi might not be exclusively your own. If there is room and another person flagging down the vehicle, then the driver will pull over and take them too. For the same reason, don't be surprised if you climb into a taxi to find someone else already sitting there; and don't feel that you can't signal for a taxi, if it is already carrying a fare. This might extend your journey slightly, if the other passenger is dropped off first. If you can't live with this, then you would be better off booking private ground transportation (see links at the bottom).
It is possible to hire a taxi for the day or part of it. For example, if you wished to travel to Chichén Itzá and stay there for 3 hours, then return to Cancún, then this could be worked out with the taxi driver. (S)he will wait at the site, so you are hiring the taxi for the entire duration. This can sometimes be cheaper and less hassle than renting a hire car; but allows you the freedom to devise your own schedule. Naturally, if you intend to do this, then you must negotiate with the taxi driver before you set off, so that there are no misunderstandings regarding costs and time. For long journeys, it is permissable to haggle the price. For short journeys, don't.
If you do not speak Spanish, then write down the address of your destination and show this to the driver. That avoids you being taken to somewhere else, because the driver misinterpreted your attempts at Spanish pronunication. However, many drivers speak perfect English and all of them will be used to driving tourists around.
Private Shuttle/Ground Transportation
The advantage of a private shuttle is that there will only be your party on it. There will be no stops between airport and hotel, unless you request one, and so the whole journey will be quick and efficient. It can save around an hour of time because of this. Most private shuttles are also air-conditioned, which is a bonus in the Cancún heat.
There will be some shuttles available for hire at the airport, but it's always a good idea to book one in advance. Many of those offering it at the airport will be timeshare representatives in disguise. You do not need to pay when you make the reservation, as that will be sorted out when the transfer takes place. However, do ask how you will recognise your driver. Most will stand at the airport entrance with your name on a card, but it is better to check in advance.
Shuttles cost between $60-$65 for a private passenger van carrying 4-7 people. This is for a roundtrip, so you will have the same van taking you back again at the end of your holiday. Some companies will ask for a portion of that during the first journey, then the remainder after you've been dropped off on the return leg. Some companies will want the entire fare paid up front. It is best to ask which is expected as you book.
Most drivers are happy to stop off en route, for example at a liquor store, if you ask them to. There will be no extra charge for this.
However, it is polite to tip your driver, particularly if they've helped with your luggage, been helpful with local information or detoured to a store for you. 20-40 pesos ($1-2 USD) is the standard rate.