Hi Jessica
Sounds fantastic - good luck with the new venture!
I've worked in Marketing for a number of different companies over the years and the ways that you can market your company hinge on how much you want to spend and how much time you want to put into it. Initially (and fairly cheaply) you could start by getting branded t-shirts/tops to wear when you are out and about. LOADS of people are going to see you on your walks so if you have your company name, phone number or website displayed well on the back then that is completely free advertising!
If you haven't got a website, then get one. You can design one yourself using a provider such as Squarespace - it is easy to do and cheap to maintain. The important bit about your website would be to have a contact form included so people can contact you if they don't have time to call you or you miss their call for some reason. The other good thing would be to have people's reviews on there - prospective new clients will always want to check you out.
Offering introductory offers or a 'refer a friend' scheme will work really well for getting new clients as everyone wants to save some cash! So offering a 10% discount to newbies or a free walk per week for existing clients if they refer one of their friends to you would be beneficial.
Good old fashioned leaflet dropping would also be something you could do easily when you are out walking your pets, and you can design these quite easily on Vistaprint or somewhere similar. Drop them through your neighbour's doors, ask if local coffee shops/pubs would put some on their notice boards/counter tops etc, ask mates to take them into their workplaces.
Overall with your marketing, find a message that will resonate with people and you can't go wrong. At this time of year maybe it is the message that you can 'brave the cold so they don't have to' (encouraging people to let you walk their dog for them on these chilly mornings) or 'ease those stressful vet visits' (by you doing it for them) or 'give your pup a Christmas haircut' (self-explanatory!). Have a think about the pain-spots that people have with their pets and put yourself forward as the one to solve them! Spread the word!!
Have you thought of printing gift vouchers at this time of year? I know for one I would love to receive a voucher that could go towards giving my pets a bit of a pampering! Just a thought....
Good luck!