First you should find a good hosting service. There is a lot of free ones. I use cba.pl hosting, but it seems it has only polish interface. However i found one that is in english and seems promising
http://freezoka.net/ . Still never tested, so you may find something better. Just try google 'free web hosting'. When you find a good host they usually have also pre-instaled packets like phpBB3 and others. You just chose one, install online, configure and there you are. Now you just find a nice theme for the look, e.g. here
http://www.phpbb3styles.net/. That's pretty much it in technical point of view.
You should also know what's the difference between html and php, and there's a huge one. That's why you got some strange results when using dreamwaver. In short php page is generated on the server side, and it's converted on-the-fly to html language. Thus you need server that interprets php language to html in order to see results of you editing. You can set server on your computer or just upload files to your remote server and then see what you got.
So it's not that simple if you want to edit forum manually, but it's very likely you don't have to, just use a nice theme that fits your purposes. Anyway if you want to get to know more about PHP language here's a decent tutorial >>
http://www.w3schools.com/php/default.asp.
Good luck!