How do I find out which gateway a Linux system is currently using?

Run

route -nee

The output will look like this:

Kernel IP routing table
Destination     Gateway         Genmask         Flags Metric Ref    Use Iface    MSS   Window irtt
192.168.0.0     0.0.0.0         255.255.255.0   U     0      0        0 eth0     0     0      0
0.0.0.0         192.168.0.1     0.0.0.0         UG    0      0        0 eth0     0     0      0
0.0.0.0         192.168.0.1     0.0.0.0         UG    0      0        0 eth0     0     0      0

This means that the system with the IP address 192.168.0.100 is using 192.168.0.1 as gateway.