Running into Azelf, Mesprit, and Uxie requires a bit of exploration. It also requires you to have beaten the game.
After beating the game, head on over to Route 20 with a Pokemon that can use both Surf and Waterfall. From the Virbank City side of Route 20, surf across the water on the southern part of the route and then head west until you find a waterfall. Climb it, then follow the stream until you find a cave — the Cave of Being.
Inside there, you'll trigger a scene with Professor Juniper and the legendary trio of the lakes, Azelf, Mesprit, and Uxie, resulting in them running off. After you've done this, you'll be able to find the three of them in various locations across Unova, although hiding.
Azelf can be found by returning to Route 23, entering from where the path to Victory Road is. From there, head west, past a shrub that can be cut down, until you find some boulders by some steps leading up to a little platform. Take a few steps around on the platform there and Azelf will reveal itself, allowing you to fight it!
|Azelf Lv. 65|
Mesprit is a little easier to track down, although may take more time. You have to return to the Celestial Tower and climb up to the very top of it, where the bell is. Up there, you'll have to step on a certain spot somewhere in the southwest corner of the top of the tower in order for Mesprit to appear.
|Mesprit Lv. 65|
Uxie is the last of the three that can be found, requiring you to travel all the way through southern Unova until you reach Nacrene City. Take a few steps around the outside of the museum, notably to the southwest of the entrance, and you should find the spot that triggers Uxie to appear, ready for you to fight it.
|Uxie Lv. 65|