check if my ethernet controller supports gigabit Ethernet

As root, run ethtool eth0 if eth0 is the NIC in question. You'll get an output like:

Settings for eth0:
Supported ports: [ TP ]
Supported link modes:   10baseT/Half 10baseT/Full 
                        100baseT/Half 100baseT/Full 
                        1000baseT/Half 1000baseT/Full 
Supports auto-negotiation: Yes
Advertised link modes:  10baseT/Half 10baseT/Full 
                        100baseT/Half 100baseT/Full 
                        1000baseT/Half 1000baseT/Full 
Advertised auto-negotiation: Yes
Speed: 1000Mb/s
Duplex: Full
Port: Twisted Pair
PHYAD: 1
Transceiver: internal
Auto-negotiation: on
Supports Wake-on: g
Wake-on: g
Current message level: 0x000000ff (255)
Link detected: yes

Should be all the info you need.

lspci should show it like "blah blah Gigabit 10/100/1000 blah blah controller". Like that. :)