As others have already posted, getting an @companyname.com email is just a matter of registering your domain with your registrar of choice. I have been using cheap-domainregistration.com for years as a low cost choice, and have never once had a problem. Most registrars, if not all, include a free email account.
As for avoiding Outlook/Thunderbird etc., a simple gmail account can be configured to poll any POP3 or IMAP account regularly and import the mail to your gmail account. Sending, forwarding etc. can all be configured as needed so your gmail email address is not seen by your customers. I believe this would give you the "gmail experience" with a customdomain.com email address as you are looking for.
What I do not know for sure, is whether or not Google approves of business use in this fashion.