The included transport map is not very good: It only show Bus lines! Railway tracks and metro tracks are shown, but not the different lines that run on them.
There is one map that does this much better: öpnvkarte.de
The site is german, but it works anywhere in the world!
The names of places are displayed in the native language of the country you are looking at.
It is also an officially featured tile layer of OSM.
This map is already implemented on other sites, so using it should not be a problem.
Then there is also openrailwaymap.org which is very good in showing the rail infrastructure. It includes 4 different themes (infrastructure, top speeds, signaling, electrification). It is available in many different languages.