How do I go about getting the necessary (real) certificates to sign my code for both Netscape Communicator and Internet Explorer?

In order to get a real certificate to sign both Netscape JAR and Microsoft CAB you have to contact one of the so called Certificate Authorities issuing certificates ensuring the identity of Internet software publishers.

One of the most familiar authorities is: VeriSign, which allows you to get certificate via the Internet, which assures the identity of a commercial software publisher by using registered credentials and commercial database matching to verify the company's identity. This Digital ID is designed to meet the needs of commercial software publishers who distribute software electronically. You can get it under URL:

http://www.verisign.com/developer/index.html

You can get a list of other Certificate Authorities acceptable by Internet Explorer from its security database:

(Tools/Internet Options/Content/Certificates/Trusted Root Certification Authorities) – I believe that most of them have web-sites.

Correspondingly, You can get such list for Netscape Navigator from its security database using Netscape's signtool utility program.