Apart from cruises, which is not a sure system to get to Pitcairn, you can get there from Auckland in the regular supplying ship to Pitcairn. This ship stops for a while in Pitcairn to discharge goods and then continue to England. They accept a few passengers. If you want to stay in the island with a family (although I saw a hostel in Pitcairn but had no time to inquire) you need a permit which will be issued to you in Auckland. Otherwise you can just visit the island during the short time of uploading cargo in the island.
Another possibility is joining a group in Mangareva and then hire a sailing boat to Pitcairn. Mangareva is just 300 kilometres from Pitcairn. I heard that there is a tourist company in California which collect tourists for Mangareva to Pitcairn for a high price, but in Pitcairn, just in february 2006, the locals told me that this sailing boat is not sailing to Pitcairn because the lack of tourists.
Somebody told me that in the near future there will be built an airport in Pitcairn, then all the romanticism to get there by boat will evaporate. So, hurry up!
Just to give a head's up to this topic. I have not done this personally but know two who did:
The "easiest" way to get to Pitcairn is on the Claymore II freight-passenger service from Mangareva, French Polynesia. The ship is on a long term charter contract from the Pitcairn government. The journey is about 540km/330 miles, so the place isn't actually as remote as is often believed, it's just very isolated. The ship is a small freight-passenger vessel with about a dozen passengers, though it is proper accommodation with cabins, a lounge and whatnot. It takes 2 days to make the crossing (about 30 hours). There is no airport so it's that or broke.
The crossing schedule is slightly weird in that the ship makes a spurt of 3 return crossings as a shuttle service, then nothing for *THREE MONTHS*, then another spurt of 3 return crossings, and so on. This good news is it means you can spend a few days on Pitcairn without a problem, you don't need to stay for the duration.
Getting to Mangareva is simple. There are two flights a week from Papeete (Tahiti, capital of French Polynesia), one direct and one stopping in Hao atoll. The Claymore II is always synchronised so it will connect with the flights into and out of Mangareva, so you won't be stuck, though Mangareva has the usual facilities.
From Papeete Air France and Air Tahiti Nui offer inter-continental flights to CDG, LAX and others. The two carriers codeshare with each other and TGV services on the French metropole, so you can book right through from Mangareva to any destination on the French TGV network. Pretty awesome :) It only breaks down once you switch from the French/French Polynesian to British/Pitcairn organisational responsibility.
Aforementioned friends reported that Marseille to Mangareva was almost seamless, though a stop-over in Papeete to deal with jet-lag and a chance to stretch is definitely worthwhile.
To book a space on the Claymore: email@example.com