R1 should advertise 192.168.10.0/24 to R2.
R2 should advertise 10.10.10.0/24 to R1 with a specific
routing protocol (RIP, OSPF, EIGRP ...).
Then, on R1 you define a default route to the internet across the serial
line connected to the internet. (ip route 0.0.0.0 255.255.255.255 next hop address).