Its really quite simple and safe!
When you right click someone with an empty hand they will receive a trade request, and can right click on you with an empty hand to open the trading GUI. If they do not respond within a certain amount of time, the request expires.
In the GUI you will see an inventory screen split down the middle with sticks, then one green wool, one grey wool, and one red wool.
Your side of the trade is always on the left side, where you will see at the bottom a gold nugget (1c), a gold ingot (10c), and a gold block (100c). These items represent how much coin you will be GIVING to the other player.
Place any items that you are trading, to a maximum of 8, in the inventory slots on the left side.
Any items that the other person is trading will show up on the right side, with any coin amounts showing at the very top, in the form of gold items.
When you are happy with the trade, click the green wool. When both players have done such, the trade will be executed. If you do not like the trade, click the red wool to leave the trade screen.
I hope this helps!
Edit: I was able to mimic the trading screen in creative to give you some examples.
Here is the default trading screen:
In this example, I am buying 64 leather from another player for 45c (5 gold nuggets = 5c and 4 gold ingots = 40c for a total of 45c):
In this final example I am selling 32 diamonds to another player for 160c (6 gold ingots = 60c and 1 gold block = 100c for a total of 160c):
