These are the main Harrogate Cycleways.
The Nidderdale Greenway is the best cycleway in Harrogate, and goes about 4 miles to Ripley. It may eventually be extended as far as Pateley Bridge. It follows the route of a dismantled railway (the Leeds to Northallerton railway, which closed in 1969), so it's largely flat. The official opening was on 25th May 2013.
Hedgehog's Nidderdale Greenway page has a map of the route, photos and descriptions, a guide to Ripley, advice on how to get to the Nidderdale Greenway, a video, and suggestions for improvements to the route.
Read more about the Nidderdale Greenway.
The Beryl Burton cycleway is a traffic-free cycle route from Old Bilton to the Nidd at High Bridge, Knaresborough. It was developed by Sustrans, to enable cyclists to avoid using the very busy A59.
From the Harrogate end, you can access the Beryl Burton cycleway via the Nidderdale Greenway and Bilton Lane. The Hedgehog guide to the Beryl Burton cycleway has a maps, route notes, photos, suggestions as to how to extend your ride, and information about Beryl Burton and her cycling career.
Read more about the Beryl Burton cycleway.
The Yorkshire Showground Greenway was opened in Spring 2014. It takes cyclists (and pedestrians) from Hornbeam Park to the Yorkshire Showground and Sainsburys Wetherby Road.
From Sainsburys, you can cross the Wetherby Road and continue via Stonefall Park to Plumpton Park shopping area (Homebase Morrisons and more).
Hedgehog's guide to the Great Yorkshire Showground Greenway has a map, route notes, photos, and information about how to get to it from either end.
Read more about the Yorkshire Showground Greenway.
The Harland Way is a Sustrans cycle and foot path between Spofforth and Wetherby, and is theoretically phase 1 of a route between Harrogate and York, the rest of which may one day be built.
It's on the trackbed of a dismantled railway, so it's flat, and a very pleasant ride. The surface for much of the way between Spofforth and Wetherby is compacted gravel, rather than tarmac, so it's not suitable for pure road bikes with thin tyres.
The Harland Way, also known as the Wetherby Railway path, goes to Wetherby, then there's an extension east of Wetherby, as far as Thorp Arch Estate and Retail Park.
Read more about Harland Way.