The advantages..
stainless - stain less. A stainless barrel will rust. Just not as fast. They can take more bad weather and not show signs then the other barrels. Some people like the looks of them some do not.
Blued = pretty!! Nothing nicer then a good dark blued barrel. Like already said, you have to oil the inside and out. But they are every bit as accurate as a stainless, and if you care for them they last for ever.
Nickel - is nothing more then a blued barrel with a paint job. I have a couple of them. They look nice until you forget them. They also stain where there is a lot of fire. But they are a good looking barrel and like the blued, with care will last for ever.
I would match the barrel color to the rifle and what you want to do with it. If your a one week hunter, any thing will do for you. If your a hunt til you drop kind of person, a stainless might be more to your habit. Also what appeals to you.. that is the most important.