Challenge Mode is unlocked from within adventure mode, at the same time as unlocking the third area. The third area will unlock once you finish all three adventure puzzles from the second area (Tree of Delicious). The typical way to finish this is to simply play through level 6, but since the second area also has a bonus puzzle level, you can also finish this puzzle and the first two adventure levels in Delicious Tree. This trophy will come during normal progression of the game.
The opening levels are pretty straightforward and slow enough that they can pretty much be beaten simply by feeding critters without aiming for fancy combos or food chains. So if you are having issues completing a level you can focus on just passing the levels. However, if you can, its a good idea to feed Smalls in these levels so that you can make progress towards the 'World's Greatest Dad' trophy.