I've actually made several siyums in different situations, on both mishna and gemara.
First of all, prepare a speech. You want this to be good. In it, explain something about what you learned and are finishing. Tune it to the audience -- don't speak on too high or too low of a level. Pick some of your favorite parts of the seder/masechet to explain. I also recommend going into some more detail about the end of the masechet, since that's what you're finishing.
After your speech, it's time to actually read the last mishna/sugya. Read it and translate, if your audience is not necessarily fluent in Mishnaic Hebrew/Aramaric. Give some explanations so people who haven't been learning it with you can follow and understand.
After you finish reading, there's some more stuff to say, generally called the "hadran", after the opening word. Say this, out loud. When you're done with the hadran, there's kaddish. Kaddish is only said if there is a minyan and the person making the siyum is eligible to say kaddish (over bar/bat mitzvah).
You're done! Breath a sigh of relief. :) It is customary, but not required, at this point, to start the next masechet, to show your enthusiasm (assuming there is a next). Invite everyone to enjoy the seudat mitzvah, and you should also eat something. :)
To avoid stumbling on unfamiliar words, i recommend going over what you say, including the text of the siyum itself, beforehand. Just be careful not to accidentally finish the masechet while you're practicing -- i just avoid saying the last couple words.