The nearest major airports are Palm Beach International (PBI, 20 minutes north) and Fort Lauderdale-Hollywood (FLL, 35 minutes south). Miami International is an hour away in good traffic and considerably longer in bad. All three have rental cars.
The Tri-Rail commuter line stops at the Delray Beach station on Congress Avenue, west of the downtown — it connects to all three airports. Tri-Rail is cheap and reliable but slow; allow an hour to PBI and 90 minutes to FLL. Brightline, the higher-speed private train, stops at Boca Raton and West Palm Beach but does not have a Delray station.
By car, exit I-95 at Atlantic Avenue (exit 52A) and head east.