I have a website that's been up since 2007 but it has a .bz extension. I just purchased a new domain name and this time with a .net extension. To do a 301 redirect is it as simple as going to the "redirects" in my control panel and permanently redirect it from the current .bz domain to the new .net domain? I'm aware that I will need to notify Google of this at Google Search Console. But I don't want the new domain to be a subdomain so how do I do that within my hosting plan. What do I do with the DNS part? Will I have to also purchase another SSL Certificate for the new .net domain?
There are multiple ways to do this depending on your provider, and Im not certain that you need to inform Google - the 301 redirect should do that.
You will need an https certificate for your new site (if its to be https enabled) - certificates are tied to domain names.
Depending on your hosting the easiest way is often to set up a new site with the new domain and cert and then set up 301 redirects from the old site, with the DNS for the old site pointing to the existing IP and for the new site to the new IP (which can be the same or different depending on the host)
From a technical POV it is possible to do this all with a single host. Depending on the website you may be able to set up seamless redirects to a canonical name or you might need to add redirect rules to the web server config. You would then have the DNS for the old and new sites the same. The tricky/difficult part here is you will need to replace the certificate with one that knows all the variants of both domains that are in use. (You also need to ensure the server recognizes the domain names as all associated with the same site)