There are two ways commonly used to "start" the chain reaction in operating nuclear power plants.
The first way uses the fact that you have quite a bit of exposed/burned fuel that emit neutrons. These can come from a number of reaction including spontaneous fission, gamma/neutron reactions, etc. You can also get spontaneous fissions from higher actinides that have build up.
The second way is to use a neutron source. One common source is antimony-beryllium. The antimony is irradiated during the power cycle and decays emitting high energy gammas. The high energy gammas react with the beryllium to create neutrons. Note that you have to irradiate the antimony to "charge" the source.
A third option, which is usually only used in the first cycle when all the fuel is fresh, is a californium source that has a high spontaneous fission rate. The problem with the californium source is that it will "burn out" if left in the reactor during power operations.
Some utilities use a neutron source, others rely on spent fuel activity. Neutron sources have other uses also. When loading the fuel, the detectors provide an adequate signal to detectors so they can detect incorrect loading patterns.