'Teach' is one of the irregular verbs, so it's 'taught' and not 'teached.' If you want to avoid using 'have a bike' twice, you can use 'one' as a stand-in for the phrase.
'Her father taught her how to ride a bike. I didn't have a bike, but/so I wanted one too.'
'..., but/so I wanted to have one too,' is also okay, but it's less concise.