Fly to FRA and take the ICE to Cologne HBF.

“Local” train tickets are just that; local to a specific country. Europe is a continent of 44 independent nations, and you can’t possibly see all of them - or even a fraction of them all - in 4 weeks. If you’re 27 or younger, Interrail is a relatively cheap way to get around most of the continent, but with Ryanair, Easyjet, Wizzair and other LCCs banging about in great numbers and to almost every nook and cranny, they’re hard to beat on both price and time.

What you should buy depends very much on what you want to see and what your priorities are. You could take a flight to Mallorca (the unofficial 17th bundesland of Germany during the summer) and stay on the beach for a month, or you could take the French riviera in style if you’ve got the money for it. If you’re more into cities, 3 of the greatest in the world are on the continent, two of which are absolute musts (Rome and Paris). If it’s solitude and nature you’re after, and don’t mind a bit of drizzle, you can’t go much wrong exploring the fjords of Norway. If it’s food you’re after, well, there’s southern France and the Emilia-Romagna region of Italy. Or, if you’re feeling very modern, the Copenhagen area of Denmark. Fancy a spot of mountaineering and carrying a black Amex? Off to Switzerland you go. Want to know what it feels like driving 250 km/h legally? Stay in Germany and book an autobahn missile with Sixt or Enterprise.

But back to the original question: The best way to “see Europe” is flying from place to place on the dirt cheap LCCs. Trains are actually pretty expensive.