Basically, I just stick them somewhere else on the internet and just hotlink to them.
Imgur is a top favorite. It's free, no subscription required, and provides share links that already include the forum img tags. Note, its corporate political policies don't align with the general ethos of this board.
I do usually resize my pictures down to somewhere in the 1000 pixel neighborhood.
The idea is that my picture is actually hosted on imgurs site, on their hardware. All I do is make a reference to it in my APS post, so that when any users browser loads my post, it knows to go display an image from imgur.
I've also used
Postimage before