Bundling the domain with hosting is convenient and it couples two things that should be independent. Changing hosts becomes harder when your domain lives inside the account you are leaving, disputes about hosting become disputes about your address, and the free domain offered with a hosting plan frequently renews at a considerably higher rate once the promotional period ends. A domain is the one asset that must remain yours regardless of any other relationship.
Register it in your own name or your business name, never under a contractor or agency account. Whoever holds the registration controls your identity, and recovering a domain from a former provider who is unresponsive is a slow process with an uncertain outcome. If somebody else registered yours, moving it into your account is worth doing while the relationship is good.
Set three things immediately after registering. Automatic renewal, because an expired domain can be lost in a way that is expensive or impossible to reverse. Registrar lock, which prevents an unauthorised transfer. And privacy protection, so your home address and phone number are not published in a public database, which most registrars now include free.
Price is the least important consideration and the difference between providers is a few dollars a year. Choose on whether the interface is comprehensible, whether support answers, and whether transferring away is straightforward. That last one is worth checking before you commit, because a registrar that makes leaving difficult is telling you something.
Register the obvious variations if they are inexpensive: the common misspelling, the plural or singular, and the alternative extension. Point them at your main address rather than building anything on them. This is not about capturing traffic, which will be minimal. It is about preventing somebody else from registering something close enough to your name to cause confusion, which is far cheaper to prevent than to resolve.
Record the registration details somewhere other than the registrar account itself, including which email address it is attached to and where the recovery goes. Domain recovery depends on proving control of that email, and a business whose domain is registered to an address at that same domain has a circular dependency that becomes apparent at the worst moment. Use an address at a different provider for the registrar account.
Set a calendar reminder for the renewal independently of the automatic renewal, because automatic renewal fails when a card expires and the notification goes to an address nobody checks. A domain that lapses can enter a redemption period with substantial fees, and in the worst case be registered by somebody else. A yearly reminder to confirm the payment method is current takes seconds and prevents the one failure in this area that is genuinely difficult to reverse.