I toured the Picos last Setember, they are really pretty and quite compact. The weather can be unpredictable with mountains and sea very close. I had some days of warm sunshine and others of storms, fogs, etc.
There is a good crossing from Potes, up to the teleferico (cable car) at Fuentas De. You need to be at the car by 9:00am to get your bike up then you have a long and tricky descent along tracks, stony trails to the road at Sotres then some more descent to Las Arenas. Do-able on 32mm tyres.
For a longer trip you really need to head offroad.
If you want some big alpine style road passes, the Pyrenees are probably better.