I had a little trouble getting the In Time quest (Beat The Time Challenge in Under 20 Minutes.). I used the following strategy and it helped me. It's just for the beginning phase. Although it may not be the most optimal strategy it's what got me the quest.
Step 1
Buy 1 Home upgrade (the 50 gold one). (In other words get the house to Lvl 2.)
Step 2
Buy 1 Magic Tree and get it to Lvl 2.
Step 3
If the Tree spawns far away restart and repeat steps 1 and 2 until the tree spawns nearby to minimize how far the Lumberdwarf has to walk.
Step 4
Buy 1 Lumberdwarf and get it to Level 2.
Step 5
Before buying your first Gatherdwarf wait until your Lumberdwarf cuts it's first tree or is about to finish cutting it. This ensures that some Lumber will be on the field for you Gathedwarf to pick up quickly.
If you do not do this, sometimes the Gatherdwarf will walk past the Lumberdwarf that is cutting the tree. This causes Gatherdwarf to walk to the end of the field wasting a lot of time.
Step 6
Do not buy Fishingdwarf until you have about 50 fish. If you want you can even go lower like 30 fish.
Step 7
Make sure you have at least 50 gold to upgrade your Dragon once for Wave 1.
Step 8
If you feel you have enough gold to get your Dragon to Level 2 before Wave 1 enemies attack, make sure to get your Gatherdwarf to Level 2. You may need to time when you purchase this upgrade as you want to make sure there is enough Lumber on the field so that your Gatherdwarf will fill his bag and not walk to the end of the field. (This step might happen
somewhere in Step 9, depending on how you see fit.)
Step 9
Again, if you feel you have enough gold to get your Dragon to Level 2 before Wave 1 enemies attack, buy a second Magic Tree. Then get Lumberjack to Level 3, then Magic Tree to Level 3 OR get Magic Tree to Level 3 and then Lumberjack to Level 3. Be certain to keep an eye on your gold to have enough to upgrade your Dragon.
(Note: I actually was following a rule to keep Magic Tree upgraded one level above Lumberdwarf so that my Lumberdwarves would not run out of trees to chop, thus walking to the end of the field and wasting time.
Dragon Level Note
A Level 2 Dragon, meaning it was upgraded only once, can beat both Waves 1 and 2 as long as it is near full health when the enemies start attacking (this may not work on Might Mode where enemies have double health, but this will work on Time Attack and should work on Normal Mode).
House and Environment Note
In the comment section many are focusing on rushing the Environment Upgrades. They are helpful and may be necessary for fish value increases, however a major amount of gold comes from lumber and house upgrades. House upgrades also mean less down time for your dwarves.
I believe I had my environment and house equal for the first 3 or 4 levels. Then I shifted focus to environment.
Also, as noted by many, house upgrade should only go up to something like level 7 maximum.
Unit and Upgrade Note
In the beginning phases it's better to upgrade your units over buying more to minimize fish consumption. You might get a second Lumberdwarf and Gatherdwarf rather soon, but it was until my fish income was steady that I allowed my unit amount to become equal with the upgrade level.