The magnetic field moves perpendicular to the current. There is a simple trick used to gauge the direction of a magnetic field created by the current: the right hand rule. According to this rule, if the thumb of the right hand is pointed in the direction of the conventional current, the direction that the fingers curl in order to make a fist (or to wrap around the wire in question) is the direction of the magnetic field.
The shape of the magnetic field depends on the shape of the conductor. When the current travels through a wire the field is circular. The field is strongest closest to the wire, and weakens as you travel outward. A more powerful magnetic field can be achieved when the wire is coiled because it concentrates the field in a smaller space.