When making a switch to a different domain, what I personally like to do for SEO purposes is build the second site on a different domain. So if your primary domain is a .com, I would build the second site as a .me .net etc.

Instead of completely shutting down the original site, I usually do a redirect and keep the redirect until my second site gets visibility on search engines.

Also, most of my sites are built on Wordpress. I find Wordpress to have the best out of the box SEO and allows quicker visibility on Search Engines.