First you need to decide if you are going to do 1 gallon batches (to start out) which are cheaper or 5 gallon batches, which is the size most home brewers and meadmakers use. Since 5 gallon carboys are so commonly used, most recipes you find will have 5 gallon batches in mind. Of course, you can always use less ingredients for a one gallon batch.
Unless you can blow glass, you can't make one. You could make a primary fermenter (what the mead will be in when it starts) out of a plastic bucket, but eventually you will need to transfer to a glass carboy anyway.
For the whole process, you will need 2 carboys. Search the yellow pages in your area for a brew shop for a carboy. You will be glad you did, and the brew shop guys are always willing to answer any questions.
If you don't have a brew shop in your area, try buying a carboy online here
http://morebeer.com/
I recommend you do 1 gallon batches to start out. If the batch goes bad, better it be a gallon of ingredients going to waste then 5 gallons. On the other hand, many meadmakers recommend against 1 gallon batches because if you have a batch aging for 6 months and then you drink it, you will be dissapointed at how fast a gallon can go.
Personally I'm still a newbie and I've only done 1 gallon batches for learning purposes and because I don't yet have room for a couple of 5 gallon carboys.
Of course you could also take the middle ground and get a 3 gallon carboy or something.
