As an avid fan of Heroku I've used it to host many websites, from very simple ones with a low amount of visits to very complex applications with a large amount of traffic.
However if you are creating your first website with Heroku you may run into a problem. Heroku uses 'Dynos' to run your website, think of them much like a single processor. The more dynos you have the more processing power you have at your disposal. With Heroku you get 1 months worth of a single dyno use for free.
The problem comes when no one has visited your site for an hour. After an hour your dyno will shut down to save Heroku's resources. If someone then visits your site it will take 5-10 seconds to spool up the dyno and get it serving the users requests. This obviously causes a delay which as us web developers know, any unnecessary delays are unacceptable. Speed matters.