Converting to Buddhism is less a physical act or ritual (like baptism in the case of Christianity) and more a spiritual change of heart. The tenets of Buddhism generally prevent Buddhists from actively converting those of other religions, so any conversion is a very personal and simple affair.
The first step for anyone when deciding to convert to Buddhism is to make sure you truly and completely understand the principles of Buddhism. Buddhism encourages any and all people to thoroughly investigate all religions, including Buddhism. Rather than delve head-first into an official conversion, read as much literature as you can on the Buddhist religion, and make sure each and every aspect of the religion in agreeable to you, as a rational thinker. In fact, one of the Buddhist principles is not to blindly follow anyone. Of course, this then applies to something as important as binding yourself to a particular religion.
Another important aspect of converting to Buddhism is to believe in and understand the Four Noble Truths, which are the core of the belief system and the way for believers to reach enlightenment. The four truths are that life is characterized by suffering, pain, and dissatisfaction, that the origin of suffering is the carving for pleasure, existence, and non-existence, that there is a way to rid yourself of this suffering, and that to rid yourself of the suffering, you must follow the Eightfold Path. If you believe those tenets, and the corresponding Eightfold Path, you are already a Buddhist by definition.
Important aspects of converting to Buddhism include practicing compassion and meditation. This can be done in the community of like-minded Buddhists, known as the Sangha. Fellowship and enlightenment are just some of the positive results of a group meeting together for meditation and discussion. Do not feel as if you need to join a particular group, or follow a leader, in order to become Buddhist. Only do what feels right to you, and question and research each choice before you commit to it.