Midi is a current loop: you send some juice down the line and it goes through an LED at the receiving end, and that LED turns on something called an optioisolator, which is read by the processor. So basically, sending midi just means getting the Arudino to turn on an LED connected through a couple 220R resistor.
There's Arduino code available for sending midi, that part is easy. There are also standard-type schematics available. This is the best page for information:http://www.audiomulch.com/midipic/
Tom Igoe has all the schematics and, as usual, an excellent and easy-to-understand explanation of the hardware and theory on this page: http://www.tigoe.net/pcomp/stamp/stamp-midi.shtml
One optoisolator I have used in the past, and that is used in many commercial designs, is a Sharp PC900. this is a good chip because the output is TTL-level digital logic.
But here's the caveat: receiving Midi with Arduino, to my knowledge, hasn't been done yet, as you have to get the processor reading the incoming data at the right speed. tricky... If you can figure it out, you'd be a hero to the electronic music community, as they would love a $40 Midi interface that could read a dozen sensors. (edit: ladyada's MidiSense
does what you want... You might want to check it out. And it is 100% open-source design too.)
PS: welcome to the forum