This achievement is obtained by picking 10 (yes, ten) of each normal (one block high) flower, each of which must have been grown naturally from a bulb. This challenging achievement is typically one of the last achievements that a player completes, due to the extreme rarity of delphinum bulbs. Currently (March 2016), there is no master version of this achievement.
The necessary flowers are as follows. Picking the newer, taller flowers/trees doesn't count.
The flowers MUST be grown naturally from bulbs; picking a flower that you or someone else placed doesn't count.
There are no real tips to make this easier, as delphinium bulbs are only available as extremely rare loot drops from treasure chests. One way to speed up the process is to buy delphinium bulbs from other players.
Beware of purchasing any "naturally grown" flowers from other players, as there is no way to determine if the flower was truly grown from a bulb or merely placed on compost by the seller.