Well, if its a ceiling that is going to lead you to the attic, be prepared to get dirty!
I took mine off, and had a wet&dry vac and sucked out all the coal dust from the rafters as I knocked down the ceiling.
An older house, its probably plaster and lathe, which will be dirty and dusty. Try and take it down in sections, (if you have a saw-zall, then use that to cut between the rafters to take it down in chunks, easier to get rid of that way, or its just going to be even more of a mess if you just knock it down all over the place.
Put plastic on all the doorways. Plastic on the floor and move every piece of furniture out of the room! Have a fan blowing in one window, and out the other window because of the dust.
If your lucky and its gypsum board, you can take it down in large chunks. The larger the better.
Wear proper protection, dust mask, (a good one), long sleeve shirt, a knit cap and hard shoes! (remember, if you take it down, the nails should be gotten rid of as soon as possible so no one steps on them and gets hurt!). Keep kids out of the area!!
Ok, now for putting up the new ceiling, go rent one of those nifty machines from a "rent-all" place, that lifts up drywall...(They are a friend to whom ever has to do this sort of work, or just get a large bunch of guys together and have them hold the thing up while you sccrew the wall board to the ceiling...Not a plesant way to spend a day).
I dunno, last time I heard they were about 12 bucks for 4 hours or so, but it may vary in different locations...
Jesse's up to 60 cents worth of replies now and this is the best one so far.
Get into the attic and take up any insulation you find. Having the insulation fall all over you is terribly unpleasant. Cut the ceiling into sections like Jesse suggested. Then you can get on top and pound it down or get below and pull it down. If you get on top, STAY ON THE JOISTS!!! Do NOT stand or put your weight on the ceiling.
Replacing the ceiling is much easier with a machine or a couple of helpers. I have done it with neither help or a machine. I nailed a 24" to 30" 2x4 to the wall about 3/4" below the joists. I sat one end of the drywall up on top of the block and used a pre-cut stud with another 24" to 30" block nailed in a "t" to support the other end. Then I hurried the heck up and put the screws in it. This is kind of like extreme carpentry, as you are risking iminnent disaster every time you let go and grab your screw gun or cordless driver.
If you can get a helper, I'd spend my rental center money on a buck of scaffolding. You can adjust the walk board up or down to get you to the work comfortably. Then you can use it to finish the ceiling.
Have fun, be safe and watch your budget. Insulation and rentals will soak up a budget quickly.
It takes some time to get the "flavor" of a DIY place where I am new at, to sort of see what the "gist" of the mill runs like...
It tough to " break into" a place and start giving advice, because some of the "old timers" have thier opinions and I have found places that they are really offended if you offer a different point of view!
I left them after a time, as I leave a lot of places, when I feel I have given my best and taken the worst.
After a few "ahem" comments from some, on here, I thought maybe I should go elsewhere
But, I stuck around for awhile, to lend a hand where ever I can, or until I am just ask to leave (or the flamers get to be too much of a trouble and I just get fed up)...
But, I thank you for the uplifting comment
Just my two cents for what its worth, and a wee bit extra for the collection plate...