You are probably not going to use much extra bandwidth with wordpress. You will see a fairly significantly higher server load than you would with a more static site. There are ways to mitigate this with caching plugins for wordpress as well as php caching libraries you can install on your server if you have enough control.
Although not its original intention, wordpress has evolved into a significantly different beast than what it started out as. Its a great place to start if the site you are doing fits well into it base functionality and plugin ecosystem well.
There are a lot of very large sites running WP so it can handle it well, but you will definitely want to look into the different caching options if the site is going to be heavily trafficked
Some things you can run into are certain things can be harder to do if you are using wordpress and if you need custom functionality there is a learning curve to writing WP plugins.
I prefer to roll my own sites most of the time, but there are quite a few instances where WP just makes way more sense from a time/cost perspective for both myself and the client.
Hope that helps a little bit