I have seen my friends using it to their advantage, and they did not want to explain it, so could one of you tell me how to live the plaza to its full potential. ( the things like big dreams( lottery ) is the thing I am talking about.)
Festival plaza is where you can initiate trades and battles in sun and moon. Unlike, say, XY for example when you could just click over to your "C-gear" and do it in-game. You can even purchase items with a currency exclusive to festival plaza. Not sure what you mean about "using it to your advantage" though. However, if you keep talking to people and answer the right questions you will have access to more items and as your festival plaza expands there are more things that you can do... if you really want to get the most out of it!
I use Festival Plaza all the time to my advantage! Not only can you trade and battle others, you can gather Festival Coins (FC) and use them on products! You can go to the Drifloom Bouncy-House to raise one of your PKMN's stats, buy a bunch of beverages (like MooMoo milk) for FCs, have a meal to raise your Pokemon's stats, win things from lotteries, raise your PKMN's friendship, and much more! You can even dye your clothes!
TL;DR: Festival plaza will always help you (f you have enough FCs). You can gather FCs by talking to new people in your Plaza.