I'm using Meiryo UI for Japanese font in Firefox but its hard to read on some websites... Here is a picture of an example:
Is this normal? Should I be using a different font? Which font?
I think it's fairly normal. I'm assuming that's from Google ニュース, which on my OS X computer using Safari looks like this: (you may need to save the picture to get the full size)
It looks to me like Google is using MS Mincho for the font, not Meiryo, and the font size in the regions you drew boxes around is 11 pt for the gray text and 13 pt for the blue links. (The bigger links for the articles in the center are using 16 pt.)
Personally, I always have to increase the font size when trying to read Kanji. While I can handle 10-12 pt fonts in English, I typically have to go with 16-18 pt fonts in Japanese to make out enough details to read text.
I assume though, as your reading proficiency increases, it becomes easier to recognize kanji at smaller sizes based on general shape and context without having to zoom in to see every little stroke. Just like how in English, you can read words without focusing on every individual letter.