Our institution has a public IP address and now we want to create a private network using a switch which can connect to the Internet through this IP address. What class network should we use, and what would be the default gateway and DNS?
You can use any of the addressing defined by RFC 1918:
10.0.0.0/8 172.16.0.0/12 192.168.0.0/16
for your private network. As far as the size of your private network, you can use whatever size fits your needs, and you can use multiple networks if necessary.
Using a switch will not work unless it is a layer-3 switch that can do NAT. Most switches cannot NAT, and you will probably need a router to NAT your private addressing to the public Internet.
You can use any usable host address in the network for your gateway address; it is simply the network address you assign to the gateway (the host, usually a router, that knows how to forward packets destined to a different network).
Your DNS is is handled by your DNS server. Some routers can have a built-in DNS server, you can have a stand-alone DNS server, or you can use one or more of the DNS servers available on the public Internet (recommendations are off-topic here).