6 hours in Barcelona

That's seriously ambitious, especially since you'll also need to consider traffic from the airport (substantial at times). I'd narrow your choices. But if you're intent on hitting your priorities, here is what I would do: check your bags at the airport (factor time for this too). Take the Aerobus from the airport to the Placa Catalunya (about 5 euro, takes 35 minutes, at least, to get into the city). It drops you off on the east side of the Placa, in front of El Corte Ingles. Walk south about two blocks to the top of Las Ramblas, walk a few blocks to get a feel for it, and get right off it again (not worth much time on your short time frame). Cut into the Bari Gotic. Head for the cathedral, take a quick spin inside. Rick has a decent walk in his Spain book of the gothic quarter area - I'd spend an hour walking around, but if you have issues with walking, this area may disappoint you (it's a lot of walking on uneven cobbles and narrow sidewalks). Then high-tail it to find a taxi to get you to Sagrada Familia. Crowds a plenty there, even on a Thursday, so you'll want to factor a good solid hour even for a "quick" tour. Then grab the metro (2 or 3 stops I think) from there back to the Placa Catalunya to catch the Aerobus back to the airport (again, factor another half hour to get back). I wouldn't bother with the hop-on-hop off because the schedule won't be quick enough for what you want to do, unless you only want to see the outsides of things. If you're jet lagged (are you arriving from NZ that day?) the city may overwhelm, especially Las Ramblas, so keep your valuables close and your wits about you. I'm not aware that there are showers at the airport, unless you belong to an airline's club that has facilities at BCN.

CL gave a great run down of what to do. I might add that with your ambitious time frame, keep an open mind that you might not make it to all your desired destinations, so be prepared to miss them if time runs out. Otherwise, it is doable, just tight.

I just returned from Barcelona - the AeroBus is fantastic (the price is now 5.30 Euro one-way), and CL is right on with the sights you can visit from Placa Catalunya. I'd give it much more time at Sagrada Familia though - the wait just to get tickets to get in took about 45 minutes. It is fascinating, but extremely crowded (I went on a Monday at lunch). The audio guide is great, but plan on at least an hour to get through the main highlights. I'd either do the Rambla or the Barri Gottic, but both, plus Sagrada Familia, are pushing it. I would definitely shoot for Sagrada Familia. Definitely get a Metro subway map; it will take you everywhere. Showers are not available at Barcelona airport.

The 6 hours I was allowing is actually calculated from when I get into town from the airport, until I need to leave town to get back to airport so hopefully will make most of my priorities. I really appreciate people taking the time to leave these posts and now feel much more confident about my ability.
Thank you all so much. 9 more sleeps until my lift-off!