How do you find current in a compound circuit with parallel and series circuits?

Current in a parallel circuit divides between the branches, depending on the relative impedance of each branch. Kirchoff's current law, which can be used to analyze that current, states simply that the sum of the currents entering and leaving a node, properly signed, always add up to zero.

Current in a series circuit is the same at every point in the circuit. This is a consequence of Kirchoff's current law, because a node in a series circuit consists of only two conductors, and the sum of the currents at that node must be zero. By implication, then, all nodes in a series circuit must have the same current.

Not asked, but discussed due to completeness, is Kirchoff's voltage law, which states that the signed sum of the voltage drops around a series circuit always add up to zero. A consequence of this is that the voltage across elements of a parallel circuit must be the same.