Bikes in trains in Barcelona:
It's free of charge but there are limitations (more strict this year). In regional trains, for short distances, you can take your bike in the train, no problem. For longer distances only up to 3 bikes are admitted, and you have to get an authorisation and a ticket for the bike (remember, free of charge).
Foldable bikes are an exception and you can carry them with you in the train but they have to be fold before entering the train.
Bikes in buses:
Not very good news my friend. There are no buses with place for bikes and the only way that you can get a bike in a bus is to have a folded one.
Rent room direct from the owners in Barcelona -
http://barcelonatrip.ucoz.ru/ in the city centre by 25 euros/night
Routes in Barcelona:
Tibidabo, Montjuic. And a special one, though a bit tough: Garraf. This is near Barcelona towards south, at 20 minutes by car.