The key when selecting an aviation mechanic or shop to maintain your newly purchased plane is "time in type": You would not take a Cessna 172 to the United Airlines maintenance facility at JFK, not would you take an A380 to the local Cessna service center (even though it would be perfectly legal to do so).
If you're buying a new aircraft (of any type) the manufacturer will be able to recommend an authorized service center for you, and you'll probably want to use an authorized service center (especially for warranty work).
If you're buying a used aircraft you can still contact the manufacturer and ask about authorized service centers. They will usually be happy to give you a list of places near you that can work on your aircraft, and who are blessed by the factory to perform warranty service, complete mandatory service bulletins where the manufacturer is supplying parts / covering labor, etc.
Aside from manufacturer recommendations there are a lot of other ways to find a maintenance shop that beat simply asking Google and picking one at random.
For the most common single-engine piston aircraft (your typical Piper Cherokee, Cessna 172/182, etc.) you can almost certainly find a mechanic at your local general aviation airport - ask the FBO where you're storing the plane for recommendations or talk to other pilots on the ramp about who they use for maintenance.
For more esoteric aircraft (high-performance singles, twins, vintage/classic/warbirds, etc.) your best resource is often a type club (e.g. The American Bonanza Society for Beechcraft Bonanzas, or ICS for the Piper Comanche) - Google can help you track down a type club, and the members can help you find mechanics who have experience working on your particular airplane.
Type clubs are also a great source of other information (like where to find parts).