There are couple of softwares in the market which can fulfill your needs. Following features are must haves in my opinion.
Sales & Marketing
Due calls management, SMS & Email alerts for Renewals, Balances, Promotions, Member Irregularity, etc.
Online Schedule Management, Automated SMS & Email Reminder to Members & Staff before the Appointment, Paid Service Appointment Management, and Session Usage Tracking
Powerful Access Control System to Track Member & Staff Attendance, Integration with Card based, Thumbprint, Palm Vein Biometric Machines, etc.
Prospect, Member, Referral, and Trial Management, Membership Upgrades, Transfers, and Freezing; Tracking all Communication with Customers
Record all Monetary Actions, Integration with Access Control (E.g. Automatically disallow if the payment is not recovered from the client)
Reports & Analytics
To help you Analyze your Income, Balances, Renewals, etc.
Some non-functional questions you need to check with the company you are evaluating:
Do they provide regular trainings (considering the high attrition rate in the fitness industry)?
How good is their customer support?
Does it have regular updates, i.e., does the software add new features every month?
Are their existing clients happy?
Maybe you can get a few client references from them and cross check with those clients. That will surely help you decide its credibility.
Disclaimer: I work with the company www.fitnessforce.com which provides a club/gym management software.