I think that's FAR TOO MUCH to cram into a week, especially as neither Ryanair and Easyjet offer Miles (JOKE). Also these airlines are point-to-point so if you are late arriving, you may miss a connection and have to wait and pay again for a new onward flight on any part of your journey.

Also this is Europe, and in many countries 25 and 26 December and 1 January are public holidays which affect transportation .......... though you are not going to the UK which virtually closes down for the Christmas and New Year period!

You might like to try the following:

Madrid to Venice ........ but you will probably spend the day travelling ... I doubt if there are direct flights so you'd have to transit via a third airport.

Venice - Florence by train
Florence - Pisa - Florence by train (doable in a longish day)
Florence to Rome by train
Rome to Barcelona by air, either paying full whack with Iberia or Alitalia (if it's still flying!) or by transiting by a third airport with another airline.

And it's also darker much earlier in winter in the cities you mention .......... I believe that NYC shares the same latitude as Lisbon in Portugal, which is even further south than Rome.

You may not get a lot of time to see these cities ........... if any!

Why not try to reduce your trip and target the most important cities. For international rail travel times, I recommend www.deutschebahn.de .

At least all the cities you want to visit are in the same time zone.