Not much at all... advertise it on craigslist or the local newspaper, post flyers around the neighborhood, at the local vet offices,shelters,parks,pet stores,etc.

Buy a couple leashes in case you need a few extras..

thats about all you need.

To actually make money in this business you would need to be walking a few dogs per hour. Walking a few dogs altogether is no easy task depending on their behaviour. If you are going to do this properly then I would recommend getting a van which can cost a lot of money, Then you need insurance for the dogs you are walking, tax, vat etc etc, its tough to get into this business as there are so many around already.

Been there, done it, failed.

An actual business or a college kid walking the neighborhood dogs? If the latter, then the cost of a package of paper and ink to print up some flyer's, a few leashes and some good walking shoes.

The cost of some fliers and business cards.

pets question and answers,www.5d2d.com

I'd recommend a pet first aid class.

Place ads on Craigslist, may in your local newspaper, and you should be good to go.

all you would need is:
*posters/ ads
*leashes in varieties of sizes and strengths
*maybe some good tennis shoes