Confirmed that these are regular farm goals so there's no dateline for completing them. The only goal with a time limit is Goal 5 - Gilding The Lily (48 hours). Once you start this goal, you will have to complete it within 48 hours.
Your next goal will not pop up if you have not reached the required level for crafting the goals items. They will pop up as your farm levels up.
The Blossom Fawn is the grand prize for completing these goals. You'll need to purchase the Wildlife Den for it.
See Floral Shop in the Buildings and Crafts Guide and Flowers in Farm Activities and Features Guide for more details. The Guides are all updated.
Goal 1: Farm-Fresh Flowers - level 23
Buy Floral Shop
Buy Flower Plots
Plant a Red Tulip
Goal 2: A Tale of Tulips - level 23
Plant Red Tulips x5
Harvest Red Tulips x6
Goal 3: A Careful Container - level 23
Make Red Vase x1
Plant Blue Orchids x2
Plant Pink Lilies x2
Goal 4: Wildest Dreams - level 23
Make Wild Bouquets x2
Give Wild Bouquets x2
Note: Goal 5 will start upon completion of Goal 4. Gather the materials needed for crafting 4 Lily Candles while you are at Goal 4 and do not submit your last item for Goal 4 until you have all the materials ready.
Make Lily Candles x4
Give Lily Candles x4
Goal 6: Late Bloomer - level 30
Feed your Blossom Fawn