Adam first came into contact with the Doctor in Henry van Statten's underground bunker in the middle of Utah. Rose takes a liking to the self proclaimed genius, and when the Utah adventure is done, she convinces the Doctor to take him along in the TARDIS. The Doctor doesn't particularly like the young man, but he decides to keep Rose happy and let him on board.
However, Adam's time on the TARDIS is rather short lived. After arriving on a space station 200,000 years in the future. Adam is tempted by the prospects of advanced information and technology. He very unwisely has a computer interface port installed in his forehead, thinking he would use it to transmit information to his answering machine back home. This plan doesn't go over to well with the Doctor, and the Time Lord very deliberately delivers the young genius back to his home on earth, destroys his answering machine, and abandons him there with unsightly advanced tech in his forehead.