I feel your pain. I literally just went through this with an apartment I moved into a few months ago. Call the management immediately and tell them there are fleas. It is their responsibility to ensure that your apartment is habitable. Some leases, although rare will offer a 30 day guarantee where you can break your lease. If this is an option, I would use it. If not, the management needs to have it sprayed.
They will likely send out a pest control tech to spray your apartment. Make sure that they spray with the chemical that inhibits growth so the eggs don't hatch. You will need to vacuum every day for about a week to make sure you're picking up the eggs. Either empty the canister or replace the bag after each vacuuming so the fleas don't continue to hatch in the bag.

As far as your furniture, you can get Borax and sprinkle it over it and just let it sit for awhile it will dehydrate any fleas or eggs in the furniture. Also wash any bedding and anything that has been on the floor in hot water.

There are also electronic flea traps you can buy that attract fleas so you can check your progress. If you don't want to buy one you can also take a shallow dish and put water and a little bit of soap (not too much) to catch them. Put it under a desk lamp in a dark room and they will be drawn to it and fall in.

If it gets to the point where you cannot stay there, make sure the landlord reimburses you for the time that you had to spend not living in the apartment. If you end up needing to leave and they do not fix the problem, you can break your lease but the procedure depends on what state you're in. At that point, I would recommend discussing with an attorney about getting a constructive eviction which would allow you to break your lease because the landlord is in breach by not making it a habitable environment.

Good luck, I'm sorry you're having to go through this. It will get resolved most likely but this is just advice based on my experience with my slumlord of a landlord.