eBrnd

A lot of curves work in GIMP involved here, sky and trees on seperate layers. The sky came out a bit grainy, but I like the colors.

Most of the shots I took of this rainbow were either out of focus because the autofocus didn't work on the sky (to little contrast there), or too bright because my camera metered off the trees when I focused on them. Before I got the settings right, the rainbow was gone...

Surely it was.
But in my thinking in that moment, the problem was the lighting, since autofocus worked on the trees which are pretty much infinity. So what I did was use manual exposure plus autofocus.

In retrospect, yes, manual focus would have been better.
Thanks for pointing this out to me! I'm just getting started at photography, so hints like this are very useful for me.

Rainbows are really tricky.

You have shot with f/13 at 40mm. There your depth of field is really large, everything between 5m and the infinity mark would have been ok for focus. DOF is not an easy concept, but important to understand. (Sorry, I am in TeacherMode (TM))

At such occasions I switch my camera to bracketing. It fires a series of shots with different exposures without the need to turn a dial. The E500 must have that too.