Hi Lindsay,
You've asked an important question. All of local search hangs on NAP (business name - address - phone number). The consistency with which a business' NAP is published across the Internet is key to its ability to rank. For SABs like yours (service area businesses like plumbers, dog walkers or carpet cleaning companies) without brick-and-mortar locations, Google requires that you include your address while creating your Google Places/Google+ Local page, but that you choose the option that hides it. Google does not want the addresses of SABs to be displayed on their local products. So, despite the fact that your address will be hidden in the SERPs, it will still be listed in the backend dashboard and Google will know what it is. Most typically, for SABs, this address is their home address.
Google considers any business as being a most relevant result for its city of location. If your dog walking business is located in San Francisco, then Google considers you as a most relevant result for queries that either stem from San Francisco-based devices or use the words 'San Francisco' in the search term. Even if the the dog walker walks dogs in Oakland, Berkeley or San Jose, Google does not care about this. Google primarily cares that the dog walker is located in San Francisco and is most likely to show her in the local results only for San Francisco-related queries.
So, typically, what SABs must do is go after the local pack results for their city of location and work towards getting organic results for their other service cities. The most common strategy for this scenario is the creation of unique, strong city landing pages for each of their service cities. All of the above is standard operating procedure.
However, your scenario presents a somewhat unique problem if I am understanding your explanation correctly. Are you saying that your service area has nothing to do with your home address and that you do not offer services in your own city - only in other cities? If so, then you are in an atypical situation. If you only walk dogs in other cities and not your own, then the use of your home address as a home base for your business is not going to help, and is likely to drive business your way that you cannot fulfill. In such a case, your options would be to:
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Not attempt to make use of the basics of Local SEO; not to use your address in any form; not to get a Google Places/Google+ Local page or other citations for your business. You would need to rely solely on organic SEO in hopes of gaining some organic visibility for these other cities where you serve but are not located. You will be at a disadvantage, however, because competitors who are physically located in these other cities will almost certainly receive preferential treatment from Google.
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Get a physical location in at least one of your service cities. This must not be a P.O. box or virtual office of any kind, as this is forbidden by Google. Instead, you would need to rent a modest physical space from a company like ActivSpace.com and set up shop there so that you could proceed as usual with Local SEO.
If, however, I've misunderstood your explanation and you simply mean that you have a large service are that does include the area in which you live, then you can proceed like any SAB. You can:
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Build a strong website, with city landing pages for each of your service cities
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Create a Google Places/Google+ Local page for your city of location, using your home address but hiding it in the dashboard
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Proceed to build citations for your business
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Utilize other forms of marketing such as linkbuilding, Social Media, video marketing etc.
I hope this answer helps you see a clear path ahead!