I am needing to find out the normal vectors to the curve to impose interface conditions in fluid-structure interactions. Could anyone give me hints?

At the point of interest on the curve, find the tangent vector to the curve. Next find the (vector) radius of curvature, which is the first normal to the curve and lies in the local plane of the curve. Finally, find the second and last normal vector as the cross-product of the tangent and the radius of curvature. You can normalize all vectors to be unit vectors.

assuming your surface is implicitely defined by a regular function (sufficiently smooth) F(x,y,z) = 0, then gridient of F at each point is equal to normal vector.

to determin tangential components of basis, pass a plan from your point based on your normal, then any two non-paralel vectors in this plan are two remained local basis, of cource when they are orthogonal your based will be orthogonal.