Ron's right. A couple more things to consider:
You will need a day to recover from the flight over. The day you arrive you will be a zombie. The next day you will not be at 100% either.
If you're zooming from one city to another by train, and it's a trip of no ore than 2 or 3 hours, you can still do part of a day signtseeing. But if you're taking a flight or a long train trip, then that day is pretty much gone, sine you will spend almost all of it getting out of one place and then getting settled in to the next (it takes time to check out, get to the train/airport, wait for your train/flight, time in transit, time getting from the rail station/airpor to your hotel, checking in, freshening up, eaating, etc.).
I'd stick to Italy, and at most one other city (probably Barcelona, since it's not far from northern Italy, or maybe Paris, just because it's your first trip to Europe and it's Paris).
Also keep in mind that you will be traveling during the holiday period. That adds some charm in some places, but the weather will be wintery (difficult for sightseeing), some things will be closed or have reduced hours, and there will be a LOT of people traveling - not necessarily foreign tourists, but lots of people doing family/holiday traveling = massive crowds at airports/trains, higher fares, etc.
Hope that helps.