I think you mean that you have a Linksys 300 Series Switch, made by Cisco. These are made for small businesses and have very few features (relative to what you're looking for). They do not have firewall capabilities, throttling, nor website blocking. They also can not run "software" (at least not like you're thinking).
You would control Internet access in a small business essentially the same way as any other business.
I would recommend using an old PC running software like pfSense or another Firewall Distro. If you prefer a "hardware" solution, Cisco makes several devices, like the WSA S170 (Web Security Appliance). There are others as well, contact a reseller you trust and see what they recommend.
In any case, you seem to be in way over your head and really need a consultant to set it up properly.
Side note: We get a number of people here who try to manage people using technology; it doesn't work that way. This most commonly comes in the form of employees "screwing off" instead of working, and management thinks that by blocking certain websites the employees will go back to work. Studies have found that employees instead spend more time screwing off in other ways on average. The solution is to talk to the employees about not getting their work done, and if the problem continues to take further actions including replacing the employee.