What exactly you mean by "co-ordinate"? Since in terms of rail travel it's not a problem to visit all the mentioned destinations, because all of them are served quite extensively and almost all are the important hubs in European rail network (thus easily reachable). So at the end it's only a matter of choice in which exactly sequence to visit each place. While purely in terms of time travel/price your plan (
Rome-Florence-Venice-Munich-Salzburg-Vienna-Budapest-Krakow-Prague-Berlin-Cologne-Paris) seems ok and from the first look it's hard to spot any obvious way to improve it
What we can do next is just advise you tips on how to travel each segment the cheapest way and, if you want, suggest particular connections, but then it would be nice to know your particular preference (daytime vs. overnight, more expensive high-speed trains vs. cheaper slower trains etc.)
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