How can a father who was not married to child's mother obtain legal visitation rights?

By going to court to establish paternity and make himself the legal father. Then requesting a visitation schedule.

If you have the child's birth certificate with your name listed as father, you have certain legal rights under the law to see your child. You probably don't need a lawyer if it's a simple situation - go to court, file a petition, and good for you. Its nice to see fathers who want to be involved, I hope things work out.