Oh goodness OP, I feel for you. I've been through this and it was awful. Admittedly I was a bit hysterical about the whole thing.
Our cat was regularly flea' d with a vet product - Frontline Combi and turns out in our area fleas were becoming immune to it. So the vet gave me a new to the market spot on the name of which escapes me right now.
I bought 2 sprays, Indorex and RIP Fleas and alternated them. I bought them online as much cheaper than the vets. I used a lot.
I just systematically went through the house. I stopped the cat roaming free in the house until his spot on worked - 48 hours.
I hoovered everyday for the first week or so, alternating rooms. I sprayed all round the edges of rooms, moved all furniture I could then sprayed a big x shape on open areas of carpet. I continued the hoovering onslaught for about another 2 weeks but not as frequently.
I washed as much as I could and that meant loads of cuddly toys that DD had in her room. It was really grim. I threw out a couple of inexpensive rugs.
I found eggs in dark places and in the grooves of the wooden floor - under the sofa, under the door mat. They're sticky so I used a wet wipe to remove them quickly and put wipe and eggs/hatchlings into a lidded jam jar. Ugh.
I'm allergic to the bites which is partly why I was so neurotic about it. I took antihistamines everyday and had jeans tucked in to socks with mossie repellent sprayed on top.
As per PPs, ask the vet for the newest to the market spot on treatment as your cat is your best weapon.
That's my Hysterical Guide. One of the reasons my infestation got so bad was because our cat was flea treated with a vet product every month that became ineffective. When I started to see bites I thought they were mossie bites and fleas didn't occur to me immediately so I was late tackling it.