When buying small yellow mangoes like the ones pictured to the left, I like to pick those that are beginning to turn yellow, still firm, and smooth. They will ripen on my counter after I bring them home. This variety—I think the stores say it is Mexican—is new to me, and absolutely lovely. Its flavor is mild yet tangy, and the flesh is extremely silky and less fibrous than other kinds. When we lived in Hawaii, mangoes a lot like this would fall on our car in the bank parking lot.
When I buy the larger mangoes like in the picture below, a friend recommended that I choose one with all three colors: green, yellow, and red. I choose ones that are firm, but give just slightly when I press my finger into the skin.
One trick I figured out on my own is that when a mango gets too ripe and losing its desirable tang, you can easily brighten up the flavor again by spritzing some fresh lime juice on it.
Lastly, this is how I cut my mangoes, and is a very popular method.
Make a slice on either side of the pit. Take each large piece, and with a sharp paring knife, cut into the flesh vertically and then horizontally to break up the fibers. Flip the piece inside out so that the squares of mango splay out. Use your knife to slice off the pieces of mango. Repeat with the other side, then peel off the skin from the middle section of mango and slice off as much of the fruit from around the pit. Cut into small pieces.