Alan,
This is an easy one. If you feel the need to fart in public, the best way is to stand up, raise one leg off the gound and force it out as loudly as possible. As soon as you have done it, strike your fingers against something, in imitation of a tuning fork, pretend to listen to them and loudly proclaim "that was a b flat!". Immediately afterwards you can ask the other inhabitants of the area to rate your fart for volume, duration and smell. Keep a running score and the winner can either get a small prize, or the satisfaction of having trumped (pun intended) all other attempts.
For burping, I would suggest that large quantities of carbonated drinks are consumed. This will give you the ability to burp almost continuously. With practice you should be able to burp "Three Blind Mice" and have other people name the tune. If you are more patriotic, I would suggest "God Save the Queen" as "Land of Hope and Glory" tends to cause significant discomfort to the digestive system.
I find this robust and open approach to expelling your bodily gases will immediately show you who has a sense of humour and who doesn't. This is because, as all eight-year olds know, farting and burping are the funniest things in the whole world. In fact, developing the abilities described above will earn you the undying respect (and in some cases awe) of small children. If you are extremely gifted, they will wait outside your house and follow you around hoping for public demonstrations.
Don't be ashamed of your bodily functions. Be proud.