The fact that you are getting buzzing sounds (picking up mains hum) from everything, including the antenna, makes me think you have a power supply / grounding problem..
If you dont have a multimeter, you need to get one! Check the power connections to EACH IC.. (put the -ve lead on pin 7, and the +ve lead on pin 14.. you should see +5v on pin 14 of each)..
Then check that the -Ve is solidly connected to the Ground of whatever amplifier you are using.
When you touch pins 8 or pin 10 on U1 (the output buffer) you should NOT hear mains hum.. If you do, it means that this point is at thigh impedence, which it should not be - it should be producing a low impedence square wave - or if the circuit is faulty, it should be held High (+5V) or low (0V) at low impedence, and induced mains hum should be greatly attenuated (to a level where it is inaudible or barely audible).
If you dont get hum when you touch this point, but get hum when you touch other points, it gets more complicated .. This would indicate that theb oscillator stage has a fault, and induced mains is being 'squared' by the gates.. But I do think this is unlikely.
Hopefully you put the IC's in sockets and didnt solder them directly on the board (dont know what the kit provides) - In my opinion good turned-pin sockets are essential - They allow one to remove IC's for debugging purposes, and remove the chance of over-soldering ICs which are easily cooked to oblivion by beginners.
This is a real simple circuit - but even simple circuits can be a pain top debug without equipment.. For this sort of circuit I would say you need a minimum of a multimeter and a logic probe (low cost unit with LED's showing +,0,and pulse states).. I have assumed you dont have a 'scope.