How fast do fleas spread?

I got a 6 week kitten yesterday from my friend, i noticed a lot of black specks in her fur, i had a feeling it was fleas but i asked my friend and she said she kept her clean and the kitten doesn't go out. This morning i decided to comb the kitten and found lots of fleas crawling in her fur. So i gave her a warm bath and used baby shampoo, it got rid of the flea dirt and a few fleas, I also combed the kitten but the fleas crawl too fast! Anyway i'm taking the kitten to the vet's tommorrow morning. I wanted to know how fast the flea would spread within the one day i've had her. The kitten slept in her cat bed which i left in my living room, we also have floor boards so the fleas can't really cling onto anything, the kitten's been on her bed most of the time, i've picked her up a few times but i don't think the fleas would spread that fast onto me, could they? because i only picked her up for a few seconds. After i gave her a bath and cleaned her i also vacumed the floors and the cat bed. i'm also going to clean the floor boards with dettol. Is there anything else i should do to get rid of them completely?