First of all, since you're a web guy, going to print you will normally size fonts in points (PT) instead of pixels (PX). Pixels are generally a screen-related measurement and you can surely google the difference between these two.
Short explanation: Illustrator and Indesign will measure fonts in points (PT) by default and 99% of print professionals will keep this default setting (unless the situation requires switching to pixels).
Then, as Axel said there is no fixed rule, as you probably did for web/css where you define each font style in your css in explicitly in writing.
For print its generally a custom job and it all depends on the amount of text. For a flyer you could go lets say 16-20 points for the biggest style (namely H1) then reduce for smaller styles.
Your bodytext (the equivalent of P in css) should be between 8-10-12 points (but again feel free to experiment and adjust to your content).
Changing to outlines removes the editing capability of a text so hopefully you will use this with caution. Also changing to outlines is a "pre-press" option which will not affect your composition, so it should not influence the sizing of your fonts.