In consoles, copy and paste should work pretty much like it does in any editor. There's one exception: right now, it's not possible to copy/paste in Safari 10 on Macs. Chrome and Firefox work fine on all operating systems that we've tried.
To copy text, just select it and hit Ctrl-C (Command-C on a Mac). If the highlight marking the selection disappears, that's normal and it means it's worked.
To paste, use Ctrl-V (Command-V on a Mac).
If you're on Linux, you should also find that the middle-click paste buffer works just fine.
Pasting multiple lines works fine in both Firefox and Chrome.
Copying multiple lines works fine in Chrome, but in Firefox it sometimes ends up munging everything onto a single line when you paste it again.
How to send ^C or ^V to the console
To send the actual Ctrl-C (^C) character to the console, just hit Control-C as usual when you have nothing selected. Ctrl-C will only do a copy action when you have console text selected.
To send a Ctrl-V (^V), use Ctrl-Shift-V instead.