Kaunas public transport consists of buses, trolleybuses, and micro-buses. Trolleybuses and buses are municipal-owned. They use the same ticket. Trolleybuses serve the trunk routes and are more frequent (typically one every 10-20 minutes). Trolleybus network is limited to the districts north of Nemunas river. Buses serve the less popular routes, including suburban (some go once in 2 hours so better check the timetables).
The public transport system works from 5:30-6:00 to some 22:00-23:00. At nights there are only very limited services of special bus routes marked with letter N. The airport bus is synchronized with flight times and operates longer. There is a modern information system at the main stops where special screens show the upcoming transport and waiting times. Destination of every bus and trolleybus is written on the vehicles themselves, but take note that the Lithuanian dative case is used. Therefore ending of the word is different than you would expect; e.g. a bus going to Šilainiai borough would be marked "Į Šilainius".
The timetables and routes of Kaunas public transport are available here.
There are no public underground parking lots, but parking at the sides of the streets is both abundant and cheap by western standards. Furthermore, the downtown is relatively compact (3,5x1,5 km) and parking in the surrounding low-rise districts (Žaliakalnis, Aleksotas, southern Vilijampolė) is both free and easier.
Leaving your car in the multi-storey parking of "Akropolis" shopping mall in the New Town district (Karaliaus Mindaugo avenue) is another alternative if you don't mind exploring the downtown on foot (Old Town is 2 km away from there).