mtr
is like traceroute
and ping
combined into one tool; it's very handy for diagnosing network issues.
dig
is a great tool for playing around with and exploring DNS.
netcat
is a good tool for quickly testing out network services, and doing things like piping the output of programs or shell scripts over the network. socat
is like netcat
on steroids, with options for wrapping things in SSL/TLS, support for IPv6, SCTP, and various other protocols, etc.
You've already mentioned wireshark
, but I'll link it here for completeness; it is a great tool for analyzing network protocols. tcpdump
can also be useful for sniffing traffic, and is recommended if you expect to be in a hostile environment; because of its complexity, wireshark
fairly commonly has security holes, and must be run as root since it needs raw access to the network. Thus, it generally isn't a good idea to run wireshark
if you suspect you may be attacked.
nmap
is also a wonderful tool, but be careful of doing any kind of scanning with it too aggressively; there are a lot of networks that have intrusion detections systems that will block you if they detect a port scan. It's generally polite to only run this against systems that you should have access to or that you have permission to scan.
curl
is a good tool for testing HTTP servers, allowing you to set arbitrary headers, view headers sent from the server, download and upload files, set cookies, etc. It's a good tool for debugging HTTP problems, as well as being good for downloading files from the web (HTTP, FTP, SFTP, etc).
telnet
is a venerable tool; some people use it like netcat
(acting as a general tool for connecting to TCP based services and playing with them manually), though it's not ideal for that as certain control characters have special meaning so sending binary data over it doesn't work well.
Have fun exploring the network. Remember to do so responsibly; don't be looking through people's private data, and don't do large scale port scans of networks that aren't yours. Be aware that certain kinds of scanning may be frowned upon, even if for completely legitimate purposes (some people get upset over a couple of simple pings; it's somewhat frustrating for network researchers).
edit: Oh, and while you're looking, it will help to refer to the RFCs for the protocols that you're looking into. RFC 791 for IPv4 and RFC 2616 for IPv6. RFC 783 for TCP, and RFC 768 for UDP. RFC 1034 and 1035 for DNS (plus many others that expand on those), RFC 822 for the basic format that email and HTTP use for message headers, RFC 2616 for HTTP 1.1, RFC 2821 for SMTP, RFC 854 for telnet, RFC 1459 (and later 2810, 2811, 2812, and 2813) for IRC. There are many other protocols out there, and many updates to the protocols I've listed, so I recommend Googling to find out more information about all of the various protocols you'll be playing with.