This is more of a tip than a guide, AND WILL NOT MAKE YOU FILTHY RICH, but should prove useful to the newbies around us who's just starting up and needs some quick money for the first mount or whatever. Take note that this 'only' works on RP realms, as it's really just a better way of saying "Ni hao, gold plx".
What you need: Good grammar, bad hygien, some social skills and a bit of patience.
Whatever side you decide to play on, ALWAYS roll a female character for your begging, it's more attractive in the eyes of the fool who's going to give you some of his hard-earned sparklies.
So, in a nutshell, what you do is you roll either a human female or an orc as well as a hunter. With the hunter, run to the nearest mailbox and mail the start-up gear to your beggar. Done? Great, let's get onto the begging.
On most RP servers, the most populated place is the Cathedral in Stormwind for Alliance, and Orgrimmar for Horde. Go there and put on your beggar persona.
Look up your first RPer, preferably 70+ as they'll be more willing to net you a bit of cash for your arr-pee, as most people roleplay knights in shining armour, or Legolas, a beggar is a nice change of pace for most.
Start your conversation politely, and play on their sympathy with something like "Good sir, I'll shine your boots for a copper. I'm dying for a piece of maggot-infested bread." or "I've got no where to sleep, got a couple of coins to spare?" etc, you get the point. Always be polite and they should give you a gold.
I've never used this technique for gold-making purposes, but I've certainly made a lot from RPing. Depending on the character, they might give you the finger or give you 10 gold. Usually you can expect 1-10 gold per player.
I've gotten about 500 gold on my level 1 RP beggar.
Like I said, nothing you get rich out off, but definitely a nice penny for those of us who's just starting out and need a bit of twocents for whatever reasons.