If I want to have the person's username saved, should it be done solely as a session or a cookie?
The standard way to implement sessions is to use a cookie. The only question is if you want to store a persistent cookie so that they don't have to login again if they close their browser and then come back later.
What should I do for those who have cookies disabled? Should I not store anything and just have them log in each time?
While you can implement a session system by passing a token through the query string of every link and a hidden input in every form, this approach makes it easy to leak sessions (as people will copy/paste URLs and give them to other people) and requires more effort (especially if you ever start directing browsers to URLs using JavaScript).
Cookies are the standard way to hold state on the WWW and it is entirely reasonable for parts of a site that depend on state (such as tracking who is logged in) to only work if the user has cookies enabled.
It is a tiny minority of users who disable cookies, if they want to log in to a site, they can re-enable them.