How can I make fonts actually work well together?

Here are 5 occurrences of ‘H’ all with the exact same font-size and other styling (which is none!). The only difference is the font used. Look how much they vary in height! It’s almost impossible for me to figure out how to make serif- and monospace fonts work well together when each font is so wildly different in height. How can I ensure that I can get different fonts to mix well size-wise without forcing people on bad connections to download my specific fonts?