The first thing you need to know is: never, ever hire a removal company. In the same way as Parisians make taking a granny-style shopping trolley to the shops look far more chic than driving your car could ever be, all the best déménagements are DIY affairs.This is not just true for poor students. Even if you own a washing machine, a fridge and several children, you and your mates can do as good a job as any pros.
Pack up your stuff. As your new Parisian apartment is doubtless tiny, you will probably have a lot of things to get rid of. You can sell second-hand books to Gibert Jeune at St Michel and advertise bigger things for free on Le Bon Coin, but the easiest way to dispose of anything that isn't of much value to you is to simply put it out on the street. Within a few hours you can guarantee that it will have been taken away by a man in a van. Regardless of the fact that the van may have red-and-white warning strips down the side and a rotating light on top, this is in no way official, as you will realise when you see that in the back of the van are a couple of the driver's mates, his wife and kids and possibly a large dog, but it's quick, efficient, and at least it saves all your unwanted possessions from ending up as landfill.
If you own a car, this is probably the moment when you realise that that Opel Corsa you bought because it would be easy to park in Paris is never going to be big enough for all of your stuff. Depending on the scale of the problem, you can either hire a van or do a few return trips on the metro with a 60-litre rucksack and a giant trolly suitcase. (Do not ever attempt this at rush hour. If you're taking the RER, aim for the carriages that are designed to take bikes, as they have more space.)
At this point, you will need some extra manpower to carry all your worldly belongings. This is where you find out how many friends you really have, as anyone willing to carry your double mattress up to your 6ième étage sans ascenseur apartment is a true friend (or planning to move in the near future and is hoping you'll return the favour).
As you're packing, don't forget that it's highly unlikely that you'll be able to park right outside your new front door. Make sure everything is well protected and can be removed quickly, as you'll either have to carry it in your arms as you face the elements or dump it on the pavement as you unload the van as quickly as possible before you create gridlock by parking in a narrow one-way street with your hazard lights on. (Another option is to park in a bus stop, but you definitely want to be done unloading before the irate man from the RATP drives up.)
When carrying everything upstairs, it's a good idea to do relays, with a friend stationed on every other landing. That way you break up the going up with some going down and nobody has to carry the heaviest things too far.
And finally, when everything is finished and you are comfortably installed in your new residence, don't forget to take all your willing helpers out for drinks or dinner. After all, that's the real reason Parisians don't use removal companies: the DIY versions is just too good an excuse to have a party. With two of my good friends having moved into Paris in the past fortnight, I should know.