Motorways: 

Vilnius – Klaipėda - Nida
Kaunas - Klaipėda - Nida
Palanga - Klaipėda - Nida
 
You can reach Neringa from Klaipėda by "Smiltynės perkėla" ferries (schedule)
New Ferry Terminal, Nemuno st 8, Klaipėda
 
All vehicle transfer prices can be found here.
 
When entering the territory of Curonian Spit National Park that is administrated by Neringa municipality, the local fee for mechanical vehicles is collected in Alksnynė’s control post. Local fee is not applied for electric vehicles.

The local fee is collected at Alksnynė control post, located on the 10th kilometer of the Smiltynė - Nida road.

Local fee rates can be found here.