Two intersecting lines are always coplanar. Each line exists in many planes, but the fact that the two intersect means they share at least one plane.
The two lines will not always share all planes, though. They can be coplanar on the same horizontal plane, for example, but not be on the same vertical plane. To determine the plane the two lines share, three points are required. The point of intersection is the first point, and then one point on each line determines the plane on which the two lines are coplanar. Also, in a three-dimensional space, parallel lines are coplanar, but skew lines, which do not intersect and are not parallel, aren't.