If you've decided you wish to use a new domain name instead of your current website address for whatever reason, it's important to preserve the goodwill you've built up under your current domain.
This "goodwill" is in two main forms.
Preserving the first type is pretty simple - just let these parties know well in advance via email and again after you've made the change to the new website address.
The second is a little more complex. Search engines can be cantankerous critters and whatever rankings you've built up may be lost if a shift to a new website address isn't performed correctly.
It can also be a very time consuming exercise to let sites that are linking to you know that your domain name has changed; with little guarantee of success that they'll bother to update their links.
Having two copies of your site - under the old domain and new website address - with a message about the new location on the old is less than optimal and may cause some headaches; both administrative and otherwise.
There is a tried and tested way to seamlessly and automatically transport visitors from your old domain to your new name - it's called a 301 redirect. Basically, it's an instruction, or set of instructions, you implement in a special file hosted under your current domain name that state "this page/site has moved; the new page/site is here" - and the visitor is automatically and seamlessy redirected. This is a Google approved method.
For a very basic move of a static site (without a database) where page names, directory locations etc. haven't changed; it can be quite simple - assuming you are familiar and comfortable working with .htaccess files. Add this following line to your htaccess file, at the top.
redirect 301 / http://www.example.com/
... where example.com is the name of your new domain
If you aren't familiar with .htaccess; you'll need to study up a bit or hire someone to do the work for you. It's important to do it right, so it may be well worth the expense engaging the services of someone experienced.
Moving a database driven site such as Wordpress is a little more complex and again, you should perhaps seek out professional services to ensure the migration is seamless.
Before you redirect your site; ensure all your content is in place on your new domain and password protect it while you're getting it ready as you don't want search engines indexing or visitors using the new site until it's ready. If you're going to use 301 redirects, also ensure all internal links on your new site point to the new domain, not the old one if you use absolute instead of relative links.
Once the redirection is put into place, leave your old site live just in case - a month or so is probably a good duration; longer if you're able to. Also maintain registration of the domain name for as long as you can. Among the reasons for this is security. If you let your domain name expire and another party then grabs it, they may have access to email still coming to your domain from customers who are not aware of the change. Also, if you have any third party services with a contact still associated with the old name, there's the potential for the new registrant to gain access to those.
How to register a name: Enter your choice in the search tool and click 'GO'. If after the check the domain names search results show your choice is available, you will then have the option to proceed to purchase registration; which is a very quick and easy process - start a search and find your ideal website address now.