How would a designer design a website for inexperienced internet users (i.e. elderly people)?

Wordpress Is the simplest thing I can think of. Of course that depends on what the client wants to do when updating the site; either it be letting you know or himself doing it. Of course he will be lost if he does know coding and trying to do it on his own. If you have to do code (personally my choice) I would put big comments right above the area where they would most likely change.

I would suggest adding pretty large fonts and focusing on making it obvious where to click to complete the desired task. As many elderly people also have reduced vision etc. I would recommend focusing on contrasts and balance.

good example of this is Avast where they make it really obvious where to click, and only show you what you absolutely need to know. In this example they only show you the download button, which is where you are most likely to click.

The beauty of the site can be maintained by a simple, straightforward design. Little elements go a long way, while cluttering the page just for design's sake will make any site look bad. I second Fredrik's advice about the larger fonts and contrast, and, in addition, keep accessibility in mind with alternative text and functionality using the keyboard. A lot of aging users can't operate mice, so make sure it can be navigated by the keyboard. Wordpress is one route you can go, but coding the site surely isn't out of the question if your abilities are there.

Are you a treehouse pro member? If so you should check out the conferences in the library, in particular "Future of Web Design - London 2014" The speaker in the very first video, Welcome, talks about how they had to do just that...design for elderly people. It's very informative and interesting! Check it out if you can.