No railway inside the country, but there is a well developed bus system. Hitchhiking is also possible. Might be harder in the main valley, but easier on the smaller roads.
Andorra is an expensive country to stay in regarding accommodation, but there are some tricks. Spread across the country there are mountain refuges that you can use for free. Mostly there's no electricity, but there is a fireplace and running water from a well or source. For a complete list, see here
All across the villages and cities you'll find free wifi.