I have a site where I have recently changed the name and the domain. My old domain redirects to the new one so Google will find it through that. I was wondering if I should submit my new domain to google to maybe speed up the process or if I should just wait.
Let nature take its course. Googlebot will hop from link to link and find it within a few days. In my experience the direct submission doesn't offer any advantage.
Add your new site address in signature, Google will find you in few days. I have also changed my domain name with a 301 redirect few hours back and I am hoping Google will find my new address like that.
Just build up your backlinks and Google (as well as the other important search engines for that matter) will find your site easily enough. At least this way it saves you the task of manually submitting your site(s) and waiting as long as 3 weeks to get spidered.