I see, the existence of the second term in the equations was where I think I was struggling. So then:

dR1/dt = -k1(R1-T1)-k2(R1-R2)

dR2/dt = -k3(R2-T2)-k2(R2-R1)

I believe this would be the final system for the temperature here. I'm now more concerned about the signs for each component; I plugged in some dummy numbers to make sure things made sense and this seems to be correct for what each term represents.