Having had (and still having 3) cats over the years,can I give you some suggestions?
My Vet said this little saying to me "Comb as if you hadn't sprayed,and sparay as if you haven't combed"
Let me make this a bit clearer.
When we got fleas we througly hoovered the carpets,and they sprayed the room with a flea spray(paying especial attention to the place where the skirting meets the carpet).
We used any reccomended(by the vet) "on cat" drops,but we also goa flea comb ang every day (or more) combed the cat.If we found fleas on the comb we pushed them off of the comb into a bowl of water(where they drowned).
Only following this schedule will you break the cat~flea~cat~flea cycle.Hoovering and spraying is essential as flea larvae can live dormant for years in furnishings,carpets etc,and will awaken when they sense a source of blood(sorry).
Conversely,the adult flea only bites the cat (or you) for food,and will then (eventually) drop off lay it's eggs (wherever) and die.Of course if you haven't hoovered/sprayed it will alll happen again.
Don't worru too much,it's not difficult to get on top.
BTW,DON'T use flea collars,they are useless as they only put the flea killing agent near the cats neck and shoulders,and can (in some case) be harmful to the cat