Hi,

My starting point is Zurich
I have several questions about connections. Is there a direct train connection from Zurich to Belgrade? What is the fastest connection from Belgrade to Thessaloniki? Like in my picture (Belgrade, Sofia, Thessaloniki) or are there faster connections? How do I get from Thessaloniki to Istanbul by train or do I have to take the bus?

 

There are no rail connections at all on your given routes. Just the northern trip to Turkey on your map that avoids Serbia and Geeece is technically possible by train.

p1]

It's not possible to go by train completely from Zürich to Beograd (regular train service does end at the border station of Croatia; there are trains from the border station in Serbia to Beograd, but not between these two countries; 10km are missing). If you don't care about boring landscape, long stays at small stations and an adventurous 2 hour walk, that's the way to go. Otherwise: bus from Zagreb.

>2]

Same for Beograd to Thessaloniki (the missing link is even longer here); there's no rail connection into Greece at all (from any country). The fastest connection is by plane (1h 30min). It's possible to go by bus at all or (slower) by train to Niš and from there by bus. 

 3]

So there's no rail connection from Greece to Turkey, too. It's possible to go by bus (about 10 hours).