The Tierra del Fuego National Park is located about 12km from Ushuaia. If you don’t have your own car, you can take a taxi or a minibus from the bus terminal to the park. Information about the entrance fees in the park and further information can be found here: http://turismoushuaia.com/zonas/parque-nacional/?lang=en_US
If you travel by minibus, it can optionally drop you off at the End of the World train. The train drives slowly about 7km through the Tierra del Fuego Park. The journey takes about 1h with a stop. I did not use the train myself and therefore I cannot give you any details. But you can find more information here: https://www.trendelfindelmundo.com.ar/en After the tour the bus would pick you up again and drive further into the park.
If you don’t want to do the train tour, the bus continues and stops at the park entrance where you have to buy your ticket for the park. When you buy your ticket you will also get a map for the park. The bus will then take you to the park and you can decide where you want to get off. Therefore it would be good to think about where you want to get off before. A map with the shuttle points where the bus can drop you off can be found here: http://turismoushuaia.com/wp-content/uploads/2018/02/EDICI%C3%93N-MAPA-PNTDF-ENG.pdf
At the tourist centre of the Tierra del Fuego National Park you can (should) register when you go hiking. After the tour you should hand in the note section back. If you go missing, someone will be notified. But the trails are well marked, so that nobody should get lost. On the map of the park which I linked, you will also find the different hiking trails.