Are there any websites that explain how a person could learn the language
and simple meaning of groups (symmetry) for easy understanding,
maybe an online mathematics game or something?
Now, as an answer to the question, I do not believe a resource in the form of a game exists (although perhaps a google search would throw one up?.........). However, there are many intriguing beginner books in this area!
As a text book, I can (personally) recommend a book entitled `Abstract Algebra: Groups, Rings and Fields' by Perry. It gives a nice overview to the whole area of abstract algebra, not just to groups.
If you are interested in groups as a way of looking at symmetry, this book looks interesting. It deals with group actions early on, and group actions are the things you are wanting to look at - a group acts on a set and it is these actions that are the symmetry that sort of `define' groups. It's what makes them interesting (in some peoples opinions, anyway).
You are correct in saying the OP needed a more detailed kick start, but my response may have done the trick? The poster may be relatively new to the world wide web and not considered a simple google search. The OP is clearly at the beginning of his/her journey into group theory and google will return lots of great papers and Youtube clips introducing the topic at minimal cost.
If the poster asked a more direct question on the topic i.e. "prove a given set is a group and commutes or is closed under a certain operation etc, etc..." a more detailed reply would've been given.
*extends olive branch*