There is a lot to be seen in Havana but it is also important to get a little outside of the city and see other areas. Fusterlandia is only about a 20 minute drive form Old Havana and one of the coolest little neighborhoods I have been to. This entire neighborhood began when artist Jose Rodriguez Fuster moved into the area and decorated his house in this fashion. Soon, other people in the neighborhood asked for his art and it slowly became an entire neighborhood! This is a must see!
Another day trip from Havana is to the wonderful town of Vinales. This is about a 2.5 hour drive from Havana which can be done in a day. The town is very cute and I would suggest staying over for a night to experience all the activities there are to offer. We went for the day and were taken to a family tobacco farm where we learned the history of tobacco farming and were shown how cigars are hand rolled.
The farmers are required to give a large percentage of their tobacco to the government for cigars to be produced in government facilities and then they are free to do whatever they want with the remainder of the tobacco. Many farmers hand roll cigars for their personal consumption.
Both trips are worth doing if you have some extra time in Havana. There are plenty of tours to book that will take you or you can hire a driver to take you where needed. Definitely make this happen!