The Specialist achievement can be earned by completing one full skill tree. This includes all the upgrades (so three points required for each skill). Once the tree is full, this achievement will unlock. As stated in the name, any tree works for this.
In terms of what's best for players, most people generally agree that if you're looking to max out one skill tree, stick with the combat one. Rogue skills are beneficial due to the added pistol damage, and the Technomancer tree has decent bonuses to injections. But neither, when viewed as a whole, outrank the combat tree in terms of usefulness.
By doing all the quests in the game, you should be able to max a tree in the third chapter, if not in the second.