Generally systems are setup so they run off timers, so in terms of operating a system you should probably just do some tests and take a good look at the system to make sure there isn't anything awry before you go away.
In terms of looking after the fish - they're tougher than you think. I've not fed my fish for weeks at a time, and they're fine. They get hungry, but they survive. So heading off down the coast for the weekend is not going to be a problem.
I personally have left my systems in the care of people who know nothing about aquaponics for several months at a time, all they've had to do is feed the fish every couple of days (so that they've grown when I get back!).
The key is to make sure your system is stable and working before you go away (e.g. check to make sure no drains are about to become clogged with roots, that your system has adequate pH buffering, your pumps still work and your water parameters look ok).
And of course the plants don't need daily (or even weekly intervention) so that side of things will be fine. Just be prepared to pull out large weeds if you go away for several months.
