The Davis-Dixon Bikeway isn't a bike path, rather a series of roads with bike lanes on them. This is probably the best way to get to Dixon.
Davis to Dixon
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Head south on Old Davis Road.
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Turn right on Tremont Road.
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Turn left on Runge Road.
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Turn right on Vaughn Street.
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You'll cross some railroad tracks and see a sign marking Dixon city limits.
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If you'd like to go to downtown Dixon, make a left on First Street.
Dixon to Davis
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Starting in downtown Dixon, head north on First Street.
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Turn right on Vaughn Street.
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Turn left on Runge Road.
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Turn right on Tremont Road.
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Turn left on Old Davis Road.
NOTE The ride to Dixon has some gradual uphill stretches. So if you're riding at about 12-13 mph, it will take you about 45 min to get from Old Davis Road & A St. to Dixon city limits. Coming back, that same stretch will take you about 35 minutes.