I'm always forgetting how to get from Manhattan to Tech Judo. Here are bus directions for future reference. (Driving directions are on the Tech Judo web site.)
You will be taking the number 127 NJ Transit bus to North Bergen. The bus schedule can be downloaded from NJ Transit. It will be about a twenty minute bus ride followed by a ten minute walk. Plan accordingly around your weigh-in time.
Go to the Port Authority bus terminal. You want to be in the building at 42nd and 8th. Follow the signs within the terminal to Gate 230. It may take you a few minutes to get oriented and find the signs if you haven't made this trip before. There are ticket machines by the gate and at other locations within the terminal. When the ticket machine prompts you for a zone, enter 2 (you can look up the zone number for North Bergen on a sign by the machine, but I thought I'd save you a moment). As of this writing, an adult round trip is $5.10.
From Port Authority the bus will go into New Jersey, weave around a bit, and get onto a main road called Tonnelle Ave. Tech Judo is on 85th St. off Tonnelle. You'll see the street numbers increasing as you proceed along Tonnelle. Get off at 83rd St. (ring for the driver to stop when you're at 81st or so). Don't worry if you miss it, you can also get off at 86th.
Walk to 85th. You'll be walking downhill on 85th for maybe the equivalent of two city blocks. You'll walk past a parking lot on your left and then the facility will be on your right. You'll probably see judoka hanging around out front. There are two entrances. Take the one on the right to sign in and get weighed.
The 127 bus doesn't run very frequently, at least not on weekends, so plan your return trip accordingly. You can catch the returning bus on Tonnelle at about 83rd or 84th, I forget exactly which.
UPDATE: I rewrote some parts and added details like the gate number and zone number for the 127 bus. Also I've learned you can take the 166 or 165 bus (thanks to referee Lillian Tom). Those buses come much more frequently, but the walk from Tech Judo to the bus stop is much longer and all uphill. I'm too lazy to write up the alternate route right now; just be aware that it exists and might come in handy.
UPDATE, 2012-04-25: Below is a map that shows the path the 127 bus takes from Port Authority to where you want to get off. You can click the image to go to Google Maps, where you can use street view, see bus alternatives, etc.