Is there such a thing as unlimited bandwidth? I'm pretty sure any host who claims that is lying, i think if your site suddenly starts taking up tons of bandwidth they'll act pretty fast to do something about it.

What kind of features do you think you'll need with the site? for example are you thinking of doing any server side programming, eg PHP, ASP, or some kind of cgi scripts? Some hosts will not allow this but may provide some cgi scripts to do common tasks, such as an emailer of web counter. Others (though I doubt many free ones) will allow you to write your own. Again you may find you need to use a database, if so you could check what software the host has (mySQL, SQL server etc) and what kind of restrictions are there for the use of the database.