Hi it depends from country to country and from train to train. 

Best countries to use trains with no reservation or with optional reservation are Norway, Denmark (except Copenhagen-Hamburg in Summer), Netherlands and Belgium (Thalys and Eurostar reservation compulsory), Germany (highspeed ICE only optional reservation), Czech Republic, Austria, Switzerland and Slovenia. 

Countries with a lot of trains with compulsory reservation, but easy to get it:

Hungary and Slovakia (only national trips, for international trips not compulsory), Italy, Poland (easy most of the trains, not easy Berlin-Poland). 

Avoid Spain and international high speed trains to/from France and currently France and Sweden. 

Check here where you can buy reservations, avoiding Interrail.eu

How to get Reservations | Community 

P.S. Most local trains do not need a reservation. Except some trains in Spain.

Add GB and IE also to the list of easy countries.

HU,SK,PL: RES only mandatory for the faster IC-trains -cost very little, can do till dep. (except at real busy times) from local machine or counter.

In general it is not really needed to print out all that stuff-as you have looked around you may have noted that most pax nowadays show it on their fone.

The nasty cases are as angelo tells: the hi-speed superfast trains in those 3 countries-but again these are so modern that e-tix are quite OK.

And as your suomenlainen: alas., the route via the 3 baltics: EE-LV-LT is not really to recommend-very few trains, bad connections, no trains over border,  a bus will work better there.

site seat61com also has lots of info on how to get there overland from UK-and back, so use sectors out of it.