SSL HandShake

I have a very basic doubt in SSL HandShake. Assume that we have a server S which uses self signed certificates. I write a Java client C which connects to the S. When C connects to S, C gets certificates from S and saves them to its truststore and the remaining part of the communication proceeds. After some time I use the same C to connect to the S, so will S send the certificates again to C, or C will use the certificates already stored in truststore. I am not good in SSL and underlying implementation of Truststore functionality in Java.

Will S send the certificates to C invariable of whether the C has certificates on its truststore?? I believe that if I have certificates in truststore C trusts S and C will not ask for certificates when I connect again?? Is my assumption right??

Is the process same for self-signed certificates and CA certificates??

Thanks in advance.