I believe ctrl+shift+i goes to whatever tab you were at last, and ctrl+shift+c goes to the element tab and enables the element selector.
I don't know if there's any other shortcuts but these are the one I use most often. You can also drag the various tabs to order them the way you like.
I think it depends on the last feature used, but F12 does that for me. You can also try ctrl+shift+i (opt+cmd+i on Mac, I think?)
I believe ctrl+shift+i goes to whatever tab you were at last, and ctrl+shift+c goes to the element tab and enables the element selector.
I don't know if there's any other shortcuts but these are the one I use most often. You can also drag the various tabs to order them the way you like.