VLANs and internet access

I work for a University research group and am trying to change the network that we use for distributed video rendering. We have our test cluster in our lab and then a cluster of ~50 computers in a public lab on the other side of the building that we use at night. During the day the lab is open to students and managed by a separate department. I need to setup a network that will allow us to use our gigabit internet in our lab, let the student lab supply there own internet access, and network them both together on gigabit for the rendering.

I will simplify it down to two computers for my question.

A--s1 -------s2--B

A is on switch 1 in my lab and B is in the student lab on switch 2. I can set up VLAN 1 for my internet on s1 and VLAN 2 for the student internet on s2. and then set A to VLANs 1,3 and B to VLANs 2,3.   When I try to connect from A to B using ip addresses how can I be sure that they will communicate on VLAN 3 and not go out through one internet connection and back in through the other one to communicate?

My last question is to make sure that STP will be fine with this setup.

Thanks,

Steven Andringa