So having a pet rat definitely has its values. The most important thing in most people's desire to have a rat is to have a friendly, curious, soft and fuzzy, furry animal to play with and hold. When it comes to trying to consider fun activities to do with your rat, there's a lot of variability, and a lot of it just depends on the rat's personality and what your interests are with your pet rat.
For the most part, people may choose to set up mini obstacle courses. They may choose to set up mazes. They may try to create an environment in a closed off room, or in a bedroom where the rat can have some out of cage time, and have a chance to scamper around, to climb, to spend time in a type of environment that enriches them as well. Some people will actually like to take their rats outside. It's recommended that they never be out loose, where there's a chance that they could fall to the ground and run off. So they do need to be placed in an appropriate type of transport carrier.
Other people like to just have their rats out for social time. They may want to just take them out on the bed and play with them. They may want to allow them to just hang out on their shoulder while they're reading email or doing some work around the house. Some rats become very adept at wanting to be shoulder rats, and actually enjoy the time that they spend allowing the owner to do their normal activities with the rat with them.
They generally aren't going to be doing a lot of tricks. Although spending time with them to socialize them, having friends come over to play with them, interacting with them, seeing how they react to certain types of new foods and treats, there all different things that you can do to enjoy being with your pet.
Most commonly, just the social time, just the interaction, just the ability to have them cuddle with you, sit with you while you're watching TV, be able to stroke or pet them, are some of the more common activities that people do with their rats.