You can't really miss the aquarium even if you're in Monterey for only a short time - but don't let the fact that it's so widely publicised put you off - it's great! Kids will love it, and I acted like a big kid for the entire day there.
It's worth allowing a whole day to wander round, although the cafe there is expensive, so take your own lunch. Free shuttle buses run from the transit plaza every 10 minute or so in the summer and drop you off right outside. The first must see is the sea otters, really amazing animals, try and catch them at feeding time, it's highly entertaining.
Other highlights (but by no means all) are the outer bay exhibit and the kelp forest tank - massive tuna whizzing around, sharks, and an elusive but incredible sunfish. The jellyfish exhibit is just mindblowing - so beautiful, very very well done. There are literally hundreds of other exhibits, full of fascinating/creepy/beautiful/just plain weird animals, and it's easy to lose track of the time here wandering about.
There's also a kids-zone too, and couple of interactive exhibits and shows. You can check out their website for some live views of the exhibits, and also prebook tickets to avoid the queues. Definitely recommended.