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Great question!
There are a couple of different ways that you can find a reliable architect, designer, or design-build firm to design your new home.
The best way to find someone that you trust is usually word-of-mouth. Talk to people that you know who have built their own home, and see what type of process they used. Ask them what they liked best and least about the process, and learn from their mistakes. If they loved their architect, design-build firm, or contractor, that's a great place to start.
Other ways to find local design solutions for your new home would include seeking out local trade associations like the AIA chapter (if you're choosing to complete this using the design-bid-build method) for your local area, or the local NARI (for other solutions like design-build, designers, or contractors) chapter for your area.
Additionally, websites like GuildQuality can point you towards local resources, as well as other social media-oriented sites like Houzz which feature projects from local professionals. Angie's List can be hit or miss, as can Yelp, although they might give you a few ideas.
Keep in mind design-build firms as an alternative to hiring just an architect!
Some states do not require registered design professionals to handle new single-family construction and renovation. If your project is relatively straight-forward or does not require extensive creative design, you could consider a residential designer - AIBD would be one way to locate those design professionals, though many excellent residential designers are not affiliated with a national trade association.
One you've determined that an architect is the right fit, begin with AIA. Many of the state American Institute of Architects affiliates list their members by areas of expertise and experience.
Possible questions for designer candidate interviews:
Good luck with your project!