You can find the list of leaders, Challengers and niche players in HRMS system in the Gartner website. The link is below for your reference. Gartner's measure of HR System providers is reliable.
There are service providers who charges per modules also. If you want recruitment, core HR and payroll module means this is complete package and will be charged accordingly. You can opt for Core HR module alone also based on your requirement.
In addition to your budget and system requirement, first identify why you want to implement the system and what are you looking out of it. AI is on high talks now. Check what the service providers view about AI technology in their system.
Another key point is, integration capacity of the system. If you are already using other systems for any purpose how flexible the new system would be for integration.
Do you have your company structure, department and function details? How about the HR processes. If you don't have prepare these things simultaneously. Employees should feel the technology in hand meaning, they should be able to apply leave from phone, should be able to Performance evaluation in system and training and so on.
Have a long term view and choose the system wisely. Ask for product demos from few vendors and choose the best.
If you need any clarification in system implementation, let me know.
27th February 2018 From India, Bengaluru