I've constructed a site and although I've submitted it to google it isn't even listed there.
That's because submitting a site to Google is a total waste of time in almost all cases. You have to remember that submitting your site yourself is like voting for yourself to Google (more or less). Yes, Google will come to your site, but why should Google care about your site if you are the only one voting for it. Get some votes from a 3rd party and Google might pay attention. In other words, I'd go straight to the directories immediately. If you have a decent budget there are a number of high quality directories that will help get you into Google. However, if you don't have a budget, there are still a number of free directories that are worth the time.
You also have to remember that if your keywords are remotely competitive, age of the links is extremely important. In other words, the sandbox is a real issue for many sites and is probably effecting yours as well.
I was wondering what would be the best: Making the new content with Joomla while not deleting the old pages until the new pages can gain a good ranking, or making the new site in Joomla using another domain I bought some time ago with this in mind?
I've faced this situation before as well. It's not an easy decision. Of course, Google likes new pages (as long as they are SEO friendly). So changing to a new system shouldn't hurt a bit. Unless your kids are going to starve, I wouldn't worry too much about making the switch. Jump in head first. I've been a little too cautious before and kept 10 or so html pages and it just made it a pain to integrate with CMS (this was before I switched to Joomla). The fact that Joomla is so much easier to update will benefit you much better in long run. Since the links coming to your site are not going anywhere, it stands to reason that your site should stay the same.
Keep in mind that you need to keep your titles the same (or better). If you remove a keyword from your title, you will lose ranking in that title. I have on many occassions.