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Is there any traffic free cycling to be had?
Not expecting much in the way of off road just to get out for an hour or 2 and not be run over.
I can put my plus wheels on if there is any action on the sand.
Old Hill bike park isn’t far away
No idea ... and depends what you want. But, I would get on Strava heatmaps and have a look in the area. The South West Coast Path might be worthwhile? We were in Ilfracombe a few weeks ago, and I just pottered about on the HT on the SW coast path. Not very tech, but nice enough and lovely views.
The cycle heatmap shows activity on footpaths so maybe not the best tool to make routes.
I’m normally pretty relaxed about riding on most footpaths, but I don’t think the SW coast path is fair game.
Bridleways - next to nothing. If you want quiet lanes then head north toward Stibb and then west toward the coast. The coast road that goes past the radar station all the way up to Welcombe and beyond is pretty quiet (single track in a lot of places, grass growing in the middle of the road type stuff). Apart from that, not an awful lot I'm afraid.
There is a couple of miles heading out of town along the canal which is pretty. It is also good to park at the inland end and cycle to town / beach avoiding the traffic on busy days.
For a day out of kid friendly cycling we've ridden the camel trail a short drive South. There is parking and a cafe at the inland end (I think it was free), also parking in Wadebridge further down. The section near Padstow gets busy. Very pretty cycling along the river and shaded on a hot day.