Like slimyguts said, you can crab right from the 12th street bridge. It will probably be slow this time of year though. Get a prime spot and keep it! You can tie your rope to the rails on the bridge and leave your traps in there overnight. I was there a few weeks ago and had mine tied up for 5 days straight. Once you leave though, someone else will get your spot, so if you find a good place in the middle, keep it as long as you're there.
The Necanicum does get a light spring run, but it is closed for angling above the 12th street bridge until 9/1. You can catch surf perch by casting right into the ocean where the Necanicum pours into the sea. I've had luck at any tide, using a pyramid weight and sandshrimp or clams as bait.
You can also catch starry flounder right in the estuary, but I haven't tried this yet. I think that the setup is pretty similar to that of surf perch, but without as much weight. Over by the high school I hear that there are some good spots. I'm not sure how the flounder eatin' is, but they make good crab bait.
And if you want to drive, there are several trout/panfish lakes within 30 minutes of downtown seaside.