Hi, 

do you take the plane to Berlin or Paris or do you go via Eurostar to mainland Europe? 

Berlin-Milan you could take the Nighttrain “Nightjet” Berlin-Zürich from there IC3 Zürich-Chur, from Chur IR train to St. Moritz and from there local trains to Tirano (Italy) and local trains Tirano-Milan. There is also the Bernina Express train Chur-Tirano.  (+ direct train from Chur, + Panoramic windows, - full with tourists, - reservation compulsory (except the carriage at the top of the train).

Instead of Milan you could end your ride also at the coast at Genova and region. From there you are very close to Cinque Terre.

To get from Milan to Budapest you will have to go in most cases via Venice (maybe you want to stop there?), From Venice there are 2 times a day Railjet trains Venice-Vienna via Semmering railway. Or you go via Trieste, Ljubljana to Budapest. 

Budapest-Paris best way is to plan to take the Nighttrain “Nightjet” Vienna-Paris (runs 3 times a week, often sold out), or with day trains. Other possibility is to go Berlin-Budapest (direct day and night trains) and from there Milan - Paris (direct trains, or you run via Switzerland or you run via the coast: Nice and Monaco). 

Swap Bud with MIl for a quicker less time eating trip. Also Overnite NJ from Wien to Mil and-via PL, from B to Bud (which city I left this morning, now just crossed Slovakia into Czech).

seat6.com - as always- has more ideas and tips about, though it mostly aims those on normal tix.

Frankly 10 days for 4 cities-that each are big tourist places is rather limited-if you also spend days out of that seeing nice landscapes-up to you, but it will eat more time.

Hello 

I assume you will be coming over from the UK by Eurostar so you can create a circular route, starting in Brussels and returning from Paris or Lille. 

Assuming you want a full day in each city 10 days is just about doable, but involves lengthy train travels.

This is how I would approach it:

Day 1 - UK-Brussels-Cologne/Bonn overnight

Day 2 Cologne to Berlin 2 nights Berlin

Day 4 Berlin to Budapest (via Vienna) 2 nights Budapest

Day 6 Budapest to Milan (via Munich or through the alps to TriesteVenice) 2 nights Milan

Day 8 Milan to Nice via Ventimiglia one night

Day 9 Nice to Paris one night

Day 10 Paris to UK.

Instead of Budapest you could consider Munich and then through the Brenner pass to Verona.

Obviously at this stage you can pencil your plans and add or erase at will.

For each leg simply use the rail planner (on line or on the rail planner app) to see frequency and times for each leg and probable reservation fees. In Germany and Austria reservations are optional but advised for busy trains, such as the one from Brussels to Cologne.

Just pick a day a few weeks in the future for examples of a journey - here is a screen shot of Berlin to Budapest.

Hello 

I assume you will be coming over from the UK by Eurostar so you can create a circular route, starting in Brussels and returning from Paris or Lille. 

Assuming you want a full day in each city 10 days is just about doable, but involves lengthy train travels.

This is how I would approach it:

Day 1 - UK-Brussels-Cologne/Bonn overnight

Day 2 Cologne to Berlin 2 nights Berlin

Day 4 Berlin to Budapest (via Vienna) 2 nights Budapest

Day 6 Budapest to Milan (via Munich or through the alps to TriesteVenice) 2 nights Milan

Day 8 Milan to Nice via Ventimiglia one night

Day 9 Nice to Paris one night

Day 10 Paris to UK.

Instead of Budapest you could consider Munich and then through the Brenner pass to Verona.

Obviously at this stage you can pencil your plans and add or erase at will.

 

For each leg simply use the rail planner (on line or on the rail planner app) to see frequency and times for each leg and probable reservation fees. In Germany and Austria reservations are optional but advised for busy trains, such as the one from Brussels to Cologne.

Just pick a day a few weeks in the future for examples of a journey - here is a screen shot of Berlin to Budapest.

 

There should be even a direct train known as EC 172 Hungaria :) 

Given these Cities 

I would go from Berlin via Prague - Vienna/Bratislava to Budapest (There is a daily direct train from Berlin via Prague and Bratislava to Budapest 

From Budapest i would either catch a train to Vienna and then the Nighttrain to Rome or Milan :) 
another but longer option would be Budapest - Ljubljana - Trieste - Venice - Milan 

From Milan you can take a direct train to Paris. 

Option B is to take a Route thorugh Switzerland 
Option C is to take a Route along the Coast (Genoa - Ventimiglia - Nice to Paris)