Taking Bike with by Aeroplane?

Hello.

Got a Question: I want to spend the Winter on La Gomera, Spain (Canarian Islands).
I live in Cologne, Germany.
And I would love to take my Bike with my (Trekking Bike).
I have no Experience, never flew.
Don´t have so much money, so I want to ask: Is it possible? What does it cost? Are there cheaper flying Companies, Alternatives. I don´t know how long I want to stay.
I found out, Return Ticket costs 300 €, but then the Return Date is fixed.

Any suggestions?

Thanks!!