  We introduce a two-band model of three-dimensional nodal line semimetals, the
Fermi surface of which at half-filling may form various one-dimensional
configurations of different topology. We study the symmetries and "drumhead"
surface states of the model, and find that the transitions between different
configurations, namely, the Lifshitz transitions, can be identified solely by
the number of gap-closing points on some high-symmetry planes in the Brillouin
zone. A global phase diagram of this model is also obtained accordingly. We
then investigate the effect of some extra terms analogous to a two-dimensional
Rashba-type spin-orbit coupling. The introduced extra terms open a gap for the
nodal line semimetals and can be useful in engineering different topological
insulating phases. We demonstrate that the behavior of surface Dirac cones in
the resulting insulating system has a clear correspondence with the different
configurations of the original nodal lines in the absence of the gap terms.
