What is George trying to accomplish by shooting Lennie before the other men can?
I think there are two possible answers to this question:
One way to look at this is as the ultimate, selfless act of friendship. George has spent many years traveling with Lennie and protecting him from a world that does not accept him. Their dream of living "off the fat of the land" and working together on their own farm is the most innocent desire to live safely and comfortably among a caring friend. By the end of the book, however, George realizes that this dream is impossible for them to achieve, especially after Lennie kills Curly's wife. His decision to kill his best friend is one that he will have to live with for the rest of his life, yet he does it as a final act of protection.
There is another way to view this, though. Another take on it is that George is finally freeing himself of the burden that Lennie has become. After this gruesome act, Lennie is dead and George is free to actually pursue his dream of owning his own land, possibly with someone else. Though George is clearly not happy after shooting Lennie, you could make an argument that all of the pair's major conflicts leading up to this event were caused by Lennie. Though he does it in the nicest way possible, by reminding Lennie of their dream, one could absolutely argue that George is making his own life easier by killing his friend.
I think it is up to the reader to decide what George's intentions were.
Of Mice and Men ends with George killing his friend Lennie. While this seems like a harsh and unforgiveable act, in reality, George is once again protecting Lennie. George knows that if Curly and the other men reach Lennie before he can, they will brutally hurt and kill him, so he sends the mob in the wrong direction, takes Carlson's gun, and goes to find Lennie himself. Instead of yelling at Lennie, George has Lennie once again recite their dream. Only then does he kill Lennie.
“Slim twitched George’s elbow. ‘Come on, George. Me an’ you’ll go an’ get a drink.’”
Slim sits beside George to try and console him. He knows George did what he had to in order to protect his friend.