How do I verify the speed of my NIC?

Just use a command like: ethtool eth0 to get the needed info.

Speed: 1000Mb/s

Use mii-tool to watch the negotiated network speed.

Ex.

eth0: no link
eth1: negotiated 100baseTx-FD, link ok

As Khaled mentioned, you should be able to run ethtool with just the interface as an argument. This will list the supported speeds, the advertised speeds, the current speed, and a bunch of other things too:

Settings for eth0:
    Supported ports: [ TP ]
    Supported link modes:   10baseT/Half 10baseT/Full 
                            100baseT/Half 100baseT/Full 
                            1000baseT/Full 
    Supports auto-negotiation: Yes
    Advertised link modes:  10baseT/Half 10baseT/Full 
                            100baseT/Half 100baseT/Full 
                            1000baseT/Full 
    Advertised auto-negotiation: Yes
    Speed: 1000Mb/s
    Duplex: Full
    Port: Twisted Pair
    PHYAD: 0
    Transceiver: internal
    Auto-negotiation: on
    Supports Wake-on: d
    Wake-on: d
    Current message level: 0x00000007 (7)
    Link detected: yes

You can also run dmesg, and grep for your interface, but this might not work if your system has been running for a long time and the current buffer no longer has that information (in that case, you'll have to grep the older /var/log/dmesg.* files):

dmesg |grep eth0
[    2.867481] e1000: eth0: e1000_probe: Intel(R) PRO/1000 Network Connection
[   19.429444] ADDRCONF(NETDEV_UP): eth0: link is not ready
[   19.431555] e1000: eth0 NIC Link is Up 1000 Mbps Full Duplex, Flow Control: None
[   19.449341] ADDRCONF(NETDEV_CHANGE): eth0: link becomes ready
[   26.972379] e1000: eth0: e1000_set_tso: TSO is Enabled
[   29.920458] eth0: no IPv6 routers present

I know this is not quite what you asked. But if you want to know the real world performance of your NIC, rather than what your computer says it should be, you can use iperf. You'll need two computers.

On computer one, run iperf in server mode:

iperf -s

On the other, run iperf in client mode:

iperf -c 192.168.0.10

If you want to see the full duplex speed, try this instead:

iperf -d -c 192.168.0.10

Substitute 192.168.0.10 for the servers IP address

Use below command

 dmesg | grep -i duplex
 Output: eth0: link up, 100Mbps, full-duplex, lpa 0x45E1

refer this