To communicate with BankID's service, you need to add our CA root certificate in your truststore to trust our server certificate. This certificate is available in "BankID Relying Party Guidelines" on this page. How you do this differs slightly depending on your environment. If this is not done you typically get various SSL/TLS errors.
For authentication to the BankID service, certificate authentication is used and your application need to use a client certificate stored in your keystore. If this is not done you typically get various SSL/TLS errors. This is issued by the same bank you purchased the service from. You don't need to install additional CA certificates to use the client certificate.
For testing, use the client certificate found on this page.