1
I buyed a new bag yesterday.
I bought a new bag yesterday.
The past simple and past participle of buy are bought, not buyed: ‘I bought these shoes last week.’ ‘I’ve bought a new phone.’
2
I bought a birthday present to my sister.
I bought a birthday present for my sister.
I bought my sister a birthday present.
Use buy something for someone or buy someone something. Do not use to for the person in this pattern.
See our grammar lesson on verbs with two objects.