Router and switches routing protocol configuration

It's basically depend upon how your network design and implemented . Assuming and consider for below options

1. if both buildings are connected with layer2 switches and one switch is connected with router in this case you don't required any routing protocols on switches . Just enable default route in router for internet connectivity and configuration sub-interface and creates VLANS for segmentation for your network and further create same VLANS in access switch and connect endpoint devices if both building ensure both switches are connected with trunk links to ensure communication of all VLANS in both switches. you can use another interface of router to connect any other networks if you wants to a access resources on another networks through point -point lease line.

2. Assuming if you are using layer 3 switches on both building and connect with router then this case you might be configured SVI in layer 3 switch then your require routing in layer 3 for forwarding traffic to router it's totally depends which routing protocols you use its depends upon size of your network . I suggest to use static router, you can use rip in case you network is interconnect with 2 or more different networks on point to point lease lines for accessing resources on different networks.