Finding an apartment in France will be one of the most difficult things you do while abroad. The easiest way to do avoid the hassle and paperwork drama is to go through an agency. However, the fees will get you if you are on a budget. Be prepared to look up to 2 months or more. You need to find an intermittent place to stay such as a hostel, a hotel, or short term apartment while you look for your permanent place.
If you are on a budget, then consider staying in a co-location, or with roommates. This will help you keep your costs down while not compromising on space. If you want to live alone, then look for a chambre de bonne, or an old servants room. These single studio rooms are often on the top floor of a building. There will be no elevator and bathrooms are in the hall way. However, you will be able to live on your own without paying a very high rent.
You can find great places to live on websites, like Pap.fr and Leboncoin.fr, that feature listings for rent by owner. Be prepared to pay up to two months rent as a deposit. Also, many renters will be asked to provide information for a guarant, or a co-signer. This is just to ensure that your rent will be payed if for any reason you cannot pay it. Co-signers need to be French and make three times the amount of the initial rent.