What is the Jewish way with regards to wedding rings? I understand that the Chatan gives the Kallah a ring under the Chupah which is the wedding ring. Can one also have an engagment ring? Can the Chatan also have a ring?

Engagement rings are fine. The Kallah may give the chatan a ring as a gift, but best not under the chuppah. The actual kiddushin should be performed with the chatan giving the only ring to the kallah.