As an Estonian E-Resident you need definitely an Estonian Domain with the suffix .ee.
Getting your domain
A complete list of service providers in Estonia gives the Estonian Internet Fondation. I selected Zone as my domain registrar.
Searching for a free domain name and paying via PayPal or credit card is qute familar. New and different is the verification you have to sign, of course,with your digital ID card plugged into your card reader.
You are done and have your first Estonian Domain. Normally it takes now some minutes to get your account information to see your new domain. You did the fist step.
Redirecting your domain
In the normal case that you already have a provider and webspace you have to redirect your domain. If the Estonian domain hoster is not offering an alias redirecting you have to change your DNS settings.
I had to open the dialog for DNS settings (see screenshot).
Then you must change the A-record of your new domain pointing to the IP-address of your existing provider. (For help on A-, CNAME-, etc. look here.)
Waiting for DNS update
The update of the DNS-Records takes a while. In my case it took roughly one hour. Then finally your first Estonian Domain is available at your new IP-address. Congratulations.