(Note: the keystrokes in this manual correspond to the Linux version's default settings. Some keystrokes may be slightly different on Cygwin or Mac OS, to better fit that platform.)
Terminator has unlimited scrollback. To clear the scrollback for the current terminal, use "Scrollback" > "Clear Scrollback", accessible from the keyboard with alt-k.
To search for a string, choose "Edit" > "Find..." from the menu, accessible with alt-f from the keyboard. If there's a selection, it's used as the search term. Regardless, you can edit the regular expression in the text field that appears at the bottom of the main window.
All matches in the current file will be highlighted, and tiny notches next to the scroll bar will show the distribution of matches throughout the file. (You can click on a notch to jump to that match.)
Most conveniently, the keys either side of 'f' can be used to rapidly scan backwards and forwards through the matches. Use alt-d to move to the previous match, and alt-g to move to the next match. These operations don't wrap, so you can tell when you've hit the first or last match. (These operations are also available from the "Find" menu, but they're only there to help you learn the keystrokes.)
You can keep a find active by dismissing the text field with return. (Pressing alt-d or alt-g also dismisses the text field while keeping the find active.)
You can cancel the find (and remove the highlighting) by dismissing the text field with escape. While a find is active, the matches will be updated as you interact with the terminal.
Terminator logs all terminal output, including escape sequences. The log files are placed, by default, in ~/.terminator/logs/. Each terminal is logged to a separate file. The file names are constructed from the date, time and the program that was originally run in the terminal. Logging can be suspended temporarily with "File" > "Show Info" > "Suspend Logging". Note, though, that terminal input is not logged, so passwords are not usually revealed. Logging can be disabled permanently, in such a way as to not generate warnings, by making the logging directory non-writable. But wait! In perhaps typical developer use, the author generates about 1 GiB of logs per year. A modern disk thus fails before the logs take up an appreciable portion. Being able to tell exactly what I typed yesterday seems well worth the space.
Like most modern terminal emulators, Terminator supports multiple tabs in each window.
You can drag tabs to rearrange them within their window. It's not currently possible to drag tabs between windows.
Double-clicking on a tab brings up the info dialog for that tab.