Whenever you have a crop ready for picking, or if you break a tool, or if someone buys something of yours at the auction, a small notification appears in the bottom-right corner of the screen. It's translucent, unobtrusive, and easy to dismiss.
However, if someone invites you to be their friend, you won't see it unless you click the 'profile' on your in game menu, view the new window, and switch to the 'home' tab and click the red icon on the left sidebar on the page.
I propose that, whenever someone adds you as a friend, a notification box appears with an [add back] and [not now] option while the player is gaming so they know instantly when they have an invite. When I first started playing, it took me more than a day to realize I was getting friend requests.