I wouldn't recommend setting up a DMZ. A DMZ network is going to open up that entire group of computers to outside network traffic which means you're going to have a lot more work cut out keeping the computers clean and protected. As bill001g is pointing out, your best option is going to be using VLANs to create multiple virtual networks. Your private, staff network would be set as one VLAN, such as 101, and your public network would be a second VLAN 102. Finally, any shared resources, such as network servers or printers, would be set in a third VLAN 103.
STAFF VLAN: 101
DEFAULT GATEWAY: 192.168.1.1
SUBNET MASK: 255.255.255.192
IP ADDRESS RANGE: 192.168.1.2 - 62
PUBLIC VLAN: 102
DEFAULT GATEWAY: 192.168.1.65
SUBNET MASK: 255.255.255.192
IP ADDRESS RANGE: 192.168.1.66 - 126
SHARED VLAN: 103
DEFAULT GATEWAY: 192.168.1.129
SUBNET MASK: 255.255.255.192
IP ADDRESS RANGE: 192.168.1.130 - 190
You will need to have a business-class router to be able to do this, as VLANs aren't supported on basic home wireless routers which are actually just gateways. One solution is dd-wrt on the right router as bill001g suggested, but another solution would be a small business firewall device like a Sonicwall TZ 105. I have set up numerous of these firewalls in different businesses locally and they have a huge range of features for the price. You will also need VLAN capable switches for connecting all devices through your switch. Then, using your firewall rules you can deny access to all traffic between the VLAN 101 and VLAN 102, but you can allow traffic between VLAN 103 and the other two, so that they each individually can have access to shared resources.
There is a lot of documentation out there for different products on how to set up VLANs and networking concepts like this, but it's not easy if it is your first time. Unless you have the ability to have your network down for a few days to try things out, test, and ensure everything is set up properly, I'd recommend that instead you find a network technician locally who can do this for you.