There are several ways to get to Montenegro. By car, by boat and most convenient by plane.

By car, you can travel along the Adriatic coast along the highway through Croatia.

By boat you can also take the ferry from Italy from Bari to Bar.
More information here:

It is certainly the fastest and most comfortable way to fly.
The main airports are Podgorica, Tivat and Dubrovnik (Croatia).

All airports are near the coast. The Dubrovnik airport is located about 14 km from the border of Montenegro.
It takes about 30 minutes from the airport to Herceg Novi, 80 minutes to Kotor, 60 minutes to Tivat.
From Tivat Airport it takes about 15 minutes to Kotor, 30 minutes to Budva and 90 minutes to Bar.

TIPP! Low cost airlines (Ryanair, Easyjet, Wizzair,etc.) offer good deals recently.