Determine your current IP address:

ifconfig

Determine your current routing table:

route -n

See which services are listening:

netstat -nlp

Ping a networked system:

ping <destination>

Trace a route:

traceroute -n <destination> # classic udp version
tcptraceroute -n <destination> <port> # tcp version on given port

If you're on Fedora/RedHat, you may need to specify -p <port>, unlike on Debian.

Do DNS lookups:

nslookup <whatever>
host <whatever>
dig <whatever>
whois <domain>

See traffic going over a network interface:

sudo tcpdump -ni eth0 # interrupt with <ctrl>-<c>

Connect to a text-oriented service using telnet. The following example illustrates an SMTP command sequence.

telnet smtp.pt.lu 25
helo pt.lu
mail from: <test1@pt.lu>
rcpt to: <test2@pt.lu>
data

Subject: test mail
To: info@nsa.gov
From: alquaida@world.net

you won't get us
.
quit

Use a system via SSH:

ssh user@<remote machine> # open a shell on the remote machine
scp user@<remote machine>:/path/to/file ./local/path/ # copies a file from remote to local
scp -r ./local/path user@<remote machine>:/path/to/ # copies a directory with contents from local to remote
ssh -X user@<remote machine> <command> # execute command on remote machine, redirect graphic display to local machine

Hint: nc or netcat is a swiss-army-knife-network tool. It can e.g. be used in many useful ways.