The Inspiration of Rassilon

Rich never gives Susan an explanation so I'll give you one. She's the Doctor's granddaughter! During his first incarnation he stole the TARDIS and escaped Gallifrey with her (reasons unknown, other family unknown.) They landed on Earth in the 60s, in Foreman's Junk Yard. His granddaughter took the alias of Susan Foreman and he just remained the Doctor. She was intrigued by human culture and entered high school, amazing her teachers with her science knowledge. Two of these, Barbara Wright and Ian Chesterton, became concerned about Susan's home life and followed her to the junkyard. They confront the Doctor about her living conditions and force themselves onto the TARDIS. While they're still boggling the Doctor decides to teach them a lesson in meddling and kidnaps them into space. His piloting skills being laughable he wasn't able to take them home so they, and Susan, became his first traveling companions. Susan eventually leaves when she falls in love with a human man of the future and Ian and Barbara make their own way back to Earth. The Doctor decides he loves seeing the Universe with others and collects more companions. And so it begins...

As for Rose, she was his first companion after the Doctor was nearly destroyed by the Time War. He was utterly traumatized by events (his own destruction of Gallifrey for one) and she helped him. They bonded deeply. Probably they became lovers. Anyway, Nine and Ten certainly love her.

Hope that helps! Any more questions just ask.