The park is reasonably safe, but robberies and worse have happened in the not too distant past, and you should keep aware of your surroundings. Definitely best to travel in groups along some of the more remote trails, especially to Templo VI.
Be especially careful with the bus rides from Flores to Tikal, as there have been a recent rash of robberies on the main highway. Only take a bus that leaves on the hour and is on time. A bus that leaves Flores late will not have the security of police protection that an on-time bus will have. Either way, do not carry more money than you will need at Tikal.
fees/permits
Adult tickets are Q150 $20US. Children under 12 are free. There are no ATMs in Tikal, so be sure to bring enough cash to cover expenses.
Tickets purchased after 3:00pm are also valid the next day. If it's possible to arrange it so that you arrive just after 3:00pm, this is the best way to experience Tikal as you can see it in the late afternoon and again the following morning. An earlier edit had said that this was not valid as of March 2011. As of July 2012, however, it is once again valid.
Also if you are a citizen you can get into Tikal for free on Sundays. This not only gives you two different times of day and thus two different experiences but also two chances to have good weather for photography.