Tryavna has quite a busy little rail station being on the main North/South line between the Gorna Orjhovitsa junction and Plachkovitsi. The station is in the modern part of town, about 10 minutes walk from the old town.
A day trip from Veliko Tarnovo makes for a pleasant hour's journey each way as you leave one mountain range for another and costs about 6 leva return.
The nearest main town to Tryavna is Gabrovo from which buses connect with the rest of the country. Buses run between Tryana and Gabrovo every half hour during the day and the bus station is next door to the train station. There's also one direct bus to and from Sofia every day, once again from the avtogara.
Tryavna is quite a small town and is a pleasure to wander. It is quite linear though following its river valley with an end-to-end distance of about two kilometres and this is served by a local bus company - "Avtospirka MNP". Bus stops are well-signposted and have route maps which show where you are and where the main sights are.
Not something I had cause to use and so no idea about timings or fares but there seemed to be quite a regular service.