There are many ways to have fun with your dog apart from taking it for walks and allowing it to play with special dog toys. For example, there are special sports for dogs like agility skill trials, dog racing(greyhound racing), hunting. This doesn't have to be a competitive activity, though some breeds really enjoy competitive activity, as in for example, border collies regularly participate in agility skill trials. Greyhounds, Siberian Huskies are suited to racing, beagles are suited to hunting.
You, as the owner will need to have considerable effort, patience, and dedication to spend time and money showing a dog in dog shows or in agility skill trials. These are very disciplined activities requiring certain skills or traits to be displayed on the part of the dog. The dog's coat also has to be flawless and up to breed standard where the dog is being shown in dog shows. This requires considerable knowledge of the characteristics of the dog's breed and expertise on the part of the dog's owner to show the dog at shows.
The bond has to be strong between you and the dog, and the dog has to be well under control in order to participate in trials, hunting and other sports. Participating in activities with your dog can help strengthen the bond between you, as well as being an enjoyable way to spend time together. You will understand your dog better from it and your dog will also understand you. It is important that both you and your dog enjoy the activity, but if a dog is not suited for an activity, this does not devalue it's other qualities.
You can also run or cycle alongside some breeds as part of your own exercise routine. However, many breeds require adapted exercise in the first year of their life. It is inadvisable to walk or play too much with the dog during it's growing phase because it's joints, bones and limbs would be too highly stressed, leading to mobility problems later in life. You can increase the distance gradually after the dog is fully grown. It's no fun either dragging around a tired puppy when you're in the middle of a run or cycle. You also will need to think about a running or cycling route for you and your dog away from other traffic, as well as the distance you both will travel together on your exercise routine. If you both take a break from running or cycling in the winter, start again in the spring, gradually increasing the distances. Stop if you dog shows signs of discomfort when exercising, and take enough water to hydrate you both, especially in warm weather. In general, dogs make great exercise companions, in that they must be exercised at least twice a day by going for a short walk, which is good for you too. Exercising dogs is great for making new friends in your community, as a lot of people tend to chat to people walking dogs while out and about.