I use the train all the time. Often visit the Lakes for the day from Hull. The Lakes are not that accessible by train. Windermere is the best bet. You can also get in from the Cumbrian Coast and Penrith.
Bike carriage on all trains is free, but you must reserve on some. On most local trains there is no reservation facility. There are some restrictions mainly in the London Area on the carriage of bikes at peak times. Your profile tells me you are from Nottingham. That would involve an East Midlands train to Sheffield or Manchester and then Transpennine Express to Windermere, most changing at Oxenholme as well. Neither of these companies require bike reservations, but space is limited so it would be advisable.
If you book on line East Coast is the only company that offer bike reservation. You can book any tickets on their site. For short journeys you can buy day returns. If you are staying away advanced purchase is your best option. Be aware that if you break your journey down into parts you can often get better fares than booking it as one through journey.
Hope that helps