Aine gives you a lot of good advice.
I will add some practical details.
The rail pass for non Europeans is called Eurail, and is good if you spend more time on the train than off it. If you are traveling 14 days and want to be in all the countries you mention, it might be worth it.
I rather cut down on countries and spend more time in the ones I'd visit.
You can find more info on the rail passes and their alternatives on the site of the man in seat sixty-one:
http://www.seat61.com/Railpass.htm
You can work out travel times with the German rail planner in English:
http://reiseauskunft.bahn.de/bin/query.e…
For traveling in a normal speed, about 2 days per city, you will need at least $US100 per person per day, on top of that will come your long distance flight, from the USA that should be about $1000, it can be much less if you search well, but more is not hard either. And your rail pass, or long distance rail tickets. See the seat 61 site for details on that. And remember that exchange rates change all the time and may be better or worse by the time you travel.
I see you have plans for 9 countries, maybe even more as Yahoo cuts out if 'words' get too long and your list of countries got cut.
To see a city you need several days, to get just a taste you need at least a few hours. And traveling from one city to the next will take at least half a day, even with overnight trains you will mostly lose half a day and can not really be in more than one country per day of travel, even when you spend every night, the whole night in a train.
And flying is not going to save you time, as an one hour flight mostly cost 5 hours, with travel to the airport and back to the center of the city.
Realistically you can see 3 countries, 5 cities and a few villages in 2 weeks, if you spend your time very economic and just travel through several countries with no more than a coffee stop when changing trains, you can do as many countries as you have on your list, but not these, as travel from Italy to Turkey to Moscow is going to cost you almost 10 days in the most economical surface travel. Flying would be faster, but not cheap at all.
And traveling that way you will only remember the insides of the trains and/or airports, so it is no use to make all those miles.
Better select 3 countries and spend 5 days in each. 3 in the capital city and 2 more in a smaller city or village. That way you will remember where you have been.
Leave the one country per day kind of travel for people trying to set a record, not for a once in a lifetime deal.
I travel often, in Europe and farther from home. Even with a slower speed of travel I do mix the different cities and only the breaks in the countryside allows me to tell them apart.