We will be on the Millenium. Do they download your digital camera photos for you on a disk? Do you have to bring your own disks or do they supply them?

They will!! Just go to the photo services counter/center. Most of the places will not let you use your own disk. IMHO it is a little pricey.
I do not know how many photos you are taking, but you might consider buying a new memory card. They are pretty cheap these days and you can use them again and again. If you run into this problem alot consider taking a laptop or getting a PSD.