(Santa Cruz de) La Palma has airport code SPC, and should not be confused with either Las Palmas (de Gran Canaria - on another Canary Island), Palma (de Mallorca - on the Baleares in the Mediterranean) or Santa Cruz (possibly more than one in the Americas).
Apart from a number of interisland flights (see below) there are five weekly direct flights La Palma - Madrid, operated by Iberia (part of the OneWorld alliance with American Airlines). You can usually connect to this flight from Europe and other continents.
There are a large number of charter flights direct to La Palma from several cities in Germany, and from Switzerland, the Netherlands, and the UK. Some of these flights can be booked as scheduled flights, in connection with intercontinental flights. Check out Lufthansa for the flights to La Palma from Germany with their daughter Condor, or KLM from the Netherlands on Transavia. The one weekly flight from the UK is operated as a charter only by JMC and can be booked through travel agents (including some on the web) in the UK.
There are frequent inter-island flights from Tenerife (north airport, TFN) to Santa Cruz de La Palma, and twice a day from Gran Canaria (LPA). BinterCanarias (a member of the Iberia group) operates these flights. Flights to La Palma should be booked in advance with the main ticket as seats are not always available in these small planes unless booked early. Flights take about 20 minutes from Tenerife and 50 minutes from Gran Canaria.
There are many scheduled and charter flights to Tenerife and Gran Canaria. Most charter flights to Tenerife operate from the south airport (TFS) and a transfer to the north airport will take at least one hour driving time, so at least about 3 hours must be accounted for. You can travel between airports by taxi (cost maybe around 60 Euros), by bus (direct or via Santa Cruz de Tenerife) or by rental car (ask for one-way to the north airport, there are often cars available at a cheap rate).