Once you complete the Trending Now Story at the Fashion Studio you will be able to complete the Rooted In Style Quest so your sims can become a hairstylist!
Watch my video walkthrough of this quest here
You firstly need to complete the Good Hair Day Quest to ‘buy and place the Stylist Career Station at the Fashion Studio.’ Go to the Fashion Studio and click on the furniture button to enter buy mode, click on collections and find A Cut Above, place the Styling Chair from this collection to complete this: 
Next you need to complete the Hair Today, Gone Tomorrow Quest by starting A Cut Above career story, this means beginning this career, click on the Styling Chair and click on A Cut Above to give your sim this career: 
Once complete the Rooted In Style Quest will begin, you will have 4 days to complete this quest to unlock 4 new hairstyles for your sims. If you don’t manage to complete in the time limit you can’t try again
There are 22 parts to complete:
1.Earn 21 simoleons (complete events until you earn enough simoleons) Reward: 25 simoleons
2.Daydream in the Styling Chair- 5 seconds, costs 3 energy Reward: 25 simoleons
3.Envision new look at a mirror (you can use a mirror at home or the one at the fashion studio)- 5 seconds, costs 3 energy Reward: 25 simoleons
4.Complete a Stylist Career event- takes up to 55 minutes to complete* (click on the stylist chair to begin) Reward: 25 simoleons
5.Take a break on a sofa- 5 seconds, costs 3 energy Reward: 25 simoleons
6.Earn 800 simoleons (complete events until you earn enough simoleons- including the Stylist Career so your sim can reach level 3) Reward: 25 simoleons and 10XP
7.Work a Long Shift in the Stylist Career- takes up to 7 hours to complete* (your sim needs to be at level 3 to unlock the long shift) Reward: 25 simoleons and 10XP
8.Poll 3 sims to find out what they look for in their hair stylist- 5 seconds, costs 3 energy each Reward: 25 simoleons
9.Complete LlamaZoom tasks and collect rewards Reward: 25 simoleons and 10XP
10.Complete 3 Long Events Reward: 25 simoleons and 10XP
11.Work a Long Shift in the Stylist Career- takes up to 7 hours to complete* Reward: 25 simoleons and 10XP
12.Complete LlamaZoom tasks and collect rewards Reward: 25 simoleons and 10XP
13.Ask 3 sims about people with hair- 5 seconds, costs 3 energy each Reward: 25 simoleons
14.Complete a Long Event Reward: 25 simoleons and 10XP
15.Complete LlamaZoom tasks and collect rewards Reward: 25 simoleons and 10XP
16.Check Simstagram for the latest trends (on a sofa)- 5 seconds, costs 3 energy Reward: 25 simoleons
17.Earn 1,000 simoleons (complete events until you earn enough simoleons) Reward: 25 simoleons and 10XP
18.Complete LlamaZoom tasks and collect rewards Reward: 25 simoleons and 10XP
19.Complete 2 Long Events Reward: 25 simoleons and 10XP
20.Change your appearance (click on the shirt button to go straight to the wardrobe/store) Reward: 25 simoleons
21.Work 2 Long Shifts in the Stylist Career- takes up to 7 hours to complete each* Reward: 25 simoleons and 10XP
22.Complete 2 LlamaZooms and collect rewards Reward: 25 simoleons and 10XP
QUEST IS COMPLETE!
*the event can be completed quicker than time shown if you complete the actions with your sims during the event
If you complete this quest in the time limit you unlock the Space Puff Pigtails Hairstyle for children sims (sorry I don’t have any children in my household at the moment so I don’t have a picture of this yet) the Braids on Braids and Retro Curls hairstyles for female adult sims and the Surfer Chic Look for male adult sims:
These are all available in 11 colours, 6 more colours are available but only if you buy them with real money:
It is a bit of a let down that we don’t unlock all the colours when we complete the quest as it is a tricky quest to complete in the time limit!
Related: Sims Mobile Main Quests
FIND THIS POST HELPFUL?
The Girl Who Games 
Your workthroughs are so helpful! If not for your work, I’d be much behind some goals in the game (I still can’t believe I went through all that time management for a few virtual haircuts 😳)
I’m glad I could help!