Please check seat61.com. The app doesn't contain timetables for the Baltic countries because the railways apparently can't be bothered to share their data with the rest of the world :/

We are in Gdansk now. How do we get to Vilnius? Looks like no trains are going that way...

Hi Monique, as @rvdborgt wrote, check seat61.com
https://www.seat61.com/Lithuania.htm

There you can find timetables etc. 
You can travel from Warsaw via Bialystok to Lithuania.

we booked some extra busses, the train connection there is really bad 

I did try that but didn't succeed. No train in the timetable. I booked a buss 

The timetables of Baltic countries are NOT coordinated.
Each country has own rail system semi-connected to the other at the borders.

It is not unusual to get from one country to the border and wait forever for the connecting train.

The new RailBaltica supposed to change that, but it remains to be seen.
Until that you should consider long-distance busses, smaller shared taxis (“marshrutka”) or renting a car.

There is a historical reason (the main rail hub is actually… St. Petersburg and Moscow!) and also technical: Baltic countries have 1520mm gauge, like (Soviet) Russia, Finland has 1524mm gauge like (Tsarist) Russia - but everything in the west has 1435mm gauge.
The 1524mm and 1520mm are close enough and through trains are possible, even at high speed. But in order to operate on 152x and 1435mm gauge either change of trains or change of boogies / wheel displacement is necessary.

The new RailBaltica supposed to have 1435mm gauge, but we have to live to see it. And in their Infinite Wisdom the “West” (AFAIK, included US) decided to build that 1435mm RailBaltica straight under the Sea to Helsinki… where the rail is (almost) the same as in Baltics and Russia - now.

While all this remains in flux, the different Baltic rail companies are unsure how to connect with each other. To make things even “better”: the service which really provided the long-distance connection there was suspended due COVID.

As a result, now for long-distance busses or private cars have to be used. Trains can be used for local service - but I doubt, any railpass would be economical.

I was quite a few times in that area, Kaliningrad / Königsberg included. Sadly, after ca. early 2000-s the best option is long-distance bus, “marshrutka” or private car. A Kiev - Vilnius train was recently introduced and it was a plan to extend it’s service until Tallin. Belarus and COVID became the problem and AFAIK it is not running now. Same with other trains, except the “Yantar’” express which is running from Kaliningrad to Moscow via Vilnius and Minsk - but that won’t help you much.

Some more technical detail: there are several solutions to “bridge” the gauge problem. For freight the most common is to re-load the cars. For passenger trains besides traditional boogie change operation there is a SUW2000 system developed by German, Polish, Russian engineers and the Talgo, developed by Spanish engineers. Both function well, but for reasons I can’t understand an entirely new - but limited rail solution was pushed and immerse amount of money wasted… without a single train running yet.

At the same time boogie change / Talgo /SUW2000 solutions operate since early 1950-s, this is how at some time the Soviet sleeping cars got as far as Madrid or HaNoi - and North Korean sleeping cars as far as Moscow. Coronavirus made things much worse of course - but the bottomless money pit called “Rail Baltica” was dug much earlier, than all of us got “blessed” by Corona.