Travel /
Train from Poland to Serbia and car from Lithuania to Budapest [15]
Allani, both routes are poor and I will tell you why. I will finally recommend a third route to you.
In Eastern Poland close to Lithuania, you have not many options. If you take the eastern way, the roads are in poor conditions, no dual carriageways. However I believe the traffic on the route might not be that heavy... Finally, you are getting into mountain driving in Subcarpathia after Rzeszów, continue through Slovakia and enter Hungary near to Miskolc. If I were in your position and had plenty of time, I would took my chances and drove
Vilnius - Bialystok-Lublin - Rzeszow-Budapest
hoping for gain from lighter traffic.
The route
Vilnius - Warszawa - Katowice - Bratislava - Budapest
is wrong in general for two reasons: Major construction works between Warszawa and Katowice; and tremendous traffic jams in Warsaw. Besides it is very long route even if after Katowice you are driving on expressways and motorways. The distance will be so long that no speed will help you recover wasted time.
My recommended route will be, pay attention now:
Segment 1: Drive from Vilnius to Ostrów Mazowiecika via Augustów, Grajewo, £omża (Routes 16 and 61 in Poland)
Segment 2: Białystok - Ostrów Mazowiecka (Route E67, National Route 8)
Segment 3: Ostrów Mazowiecka -> Mińsk Mazowiecki via £ochów (switch to Route 50 southwards)
Segment 4: Continue on Route 50 to (near to) Grójec and switch to E77, National Route 7 towards Radom
Segment 5: Drive on Route 7 via Radom, Kielce, to Kraków
Segment 6: Pass Kraków and stay on E77/7 until you reach Polish/Slovak border in Chyżne/Trstena
Segment 7: Go on Slovak roads via Dolny Kubin, Ruzomberok, Banska Bystrica, Krupina, Šahy border crossing to Hungary, continue via Vac to Budapest.
I would choose this route myself if I were you.
Note: You need to buy the sticker for your car for Slovakia and "electronic sticker" for Hungary, for paying toll-roads in advance.
Sorry! I was developing the route and made mistakes.
Here's the recommended route, correctly:
Segment 1: Drive from Vilnius to Ostrów Mazowiecka via Augustów, Grajewo, £omża (Routes 16 and 61 in Poland)
Segment 2: Ostrów Mazowiecka -> Mińsk Mazowiecki via £ochów (switch to Route 50 southwards)
Segment 3: Continue on Route 50 to (near to) Grójec and switch to E77, National Route 7 towards Radom
Segment 4: Drive on Route 7 via Radom, Kielce, to Kraków
Segment 5: Pass Kraków and stay on E77/7 until you reach Polish/Slovak border in Chyżne/Trstena
Segment 6: Go on Slovak roads via Dolny Kubin, Ruzomberok, Banska Bystrica, Krupina, Šahy border crossing to Hungary, continue via Vac to Budapest.
I have to add I live some 10 km to the Warsaw -> Katowice road and some 15 km to the E77/7, at the south side of Warsaw and was taking all possible routes to Budapest already in the past; I've been also from my place farther than Augustów towards Lithuania several times; so I know what I'm talking about. ;-)