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The official Turismo de Vigo website has a dedicated page on hiking with plenty of information. It mentions hiking routes for beginners and experienced hikers, as well as various sea-view trails.
The Federacion Gallega de Montanismo (the Galician Mountaineering Federation) is headquartered in Vigo, hence you could try contacting them for information on trails, and maybe even to find someone to hike with. Moreover the site hosts a link to a PDF document detailing routes and hiking logistics (in Spanish).