The process needs to begin at the design stages of a website -- if you're designing with tiny 9px font sizes, in shades of the lightest gray, your doing it wrong from the start.
As a UI/UX designer, font size & contrast ratio is the easiest place to start, you can use this contrast checker tool to continually test your ratios.
Although a recommended font size of 12px is advised, remember, font's with a proportionately small X height, like Futura, read smaller than screen favorite Georgia, with its very large X height.