You can put mead into a whole range of bottles, so you have a lot of options. It depends a lot on how you intend to drink it, and what kind of equipment you have.
I personally prefer to use basic beer bottles because I have a lot of them around, and I have a good capper. If you plan to carbonate your mead this will definitely be the easiest option for you but they work fine for still mead too.
However, mead is usually sold in wine bottles with corks, so you can also use those. Any color wine bottle will work, since mead is not so sensitive to light like beer is (but don't leave it in a clear bottle out on your sunny porch for months on end). When I use wine bottles I use whatever I can get my hands on for free, and size #8 1 3/4" first quality corks.
Bottling style is really mostly a question of taste. There are disagreements and discussions about what is best for aging (and I'm not going to weigh in on it because I don't know enough), but I make my choices based on what bottles I have on hand.