The best way to find out if you can modify the code is try on a site and see what happens. Obviously you don't want this to cost a lot of money.
If you go to the free hosting link at http://www.freewebhosting.comxa.com/ you get a free hosting account with a free domain at comxa.com and free setup and free site builder. (no catches, no adds, just free, free, free with lots of traffic and space as well).
Once you have set up your account, go to the members area, open c-panel and then click on the site builder button. Go through the setup steps of choosing a template and entering basic website data (name, company details etc). Then you can go into site builder and start adding stuff using the wysiwyg editor. save the page and then in a separate tab in your browser, view the newly saved page, online instantly.
Now you can use the wysiwyg editor to do things like changing font size, colour, add pictures etc, check that it shows up how you want on the actual website. Then come back to the editor and click on the 'show html' button and you can go and look directly at the html code behind the page and try making changes there directly. Once again, save the changes and then refresh the website view to see the results straight away.
Other things to try are making changes to the layout of the whole site by editing the templates (try removing the "powered by... " from the footers (it is possible and not too tricky) and signing up for and inserting google adsense ads, google search box etc.
Another thing you can do is when you are looking at the website, view the page source using the browser menu to look at the html code.