Here is the full list of all 13 Gym Empire - there are Gym Tycoon Sim Management achievements.
You followed Joe's tutorial.
You trained 5 Clients.
You trained 20 Clients.
You have 5 Members.
You have 20 Members.
You have 100 Members.
You trained 100 Clients.
Gain a Gym Popularity Level of 5
Gain a Gym Popularity Level of 20
Gain a 2 Star Reputation.
Gain a 5 Star Reputation.
Buy a second Premises.
Buy all the Premises available.