As a former Catholic who has become a fully convinced Calvinist, I'd like to offer my 2 cents worth. There has been a lot of good advice - pick a topic and stick to it (much like a thread on the forum). If you find yourselves constantly straying into another area, you may want to change the topic to the other area, as this may need to be resolved first (like a building, you don't start with the roof or walls, but you make sure you have a good foundation to build on).
Don't be dismissive, but address what your opponent says. For example, in another thread, I posted a paragraph from Origen and gave a link to my source. A Catholic dismissed it without comment by pointing out that the source was "an Orthodox website". I then posted the next paragraph in what Origen wrote, and pointed out that the previous paragraph was identical to what is found in the Faith Database, which is available for sale in the CA shop. The same Catholic disparaged Origen and accused me of surfing every fundamentalist website looking for something to use. I see very little value in carrying on a discussion with someone like that. In discussions on Purgatory, the book of Maccabees has been brought up. Rather than simply dismiss it with, "I don't consider Maccabees to be Scripture", I actually address the verses cited.
Very important - CHECK YOUR SOURCES!!! Just to point out a couple that I've heard before, no ECF that I'm aware of ever said, "Rome has spoken, the case is closed", and the source for the 33,000+ "Protestant" denominations includes (among other things) over 1,000 Catholic denominations (to be fair, only 242 are Roman Catholic). One poster mentioned how he would handle a discussion on Purgatory. I would check Catholic sources (like the CCC, Catholic Encyclopedia, church councils, etc...) to see what the Church says about the subject. Be prepared to give your sources!
Be careful about using quote books. I try not to use the ECF's because they had a wide variety of opinions. When I do use them, I will rarely give a quote, but will post the full paragraph (as well as the citation of where it can be found, in case anyone wants to look it up for themselves) in which the quote is given so that the context can be understood.
Finally, I believe you can learn from listening to others. For example, Tim Staples did a debate with a Calvinist (James White) on Purgatory, which should be available in the CA shop. James White has debated other Catholic apologists (Sungenis, Pacwa, and Stravinskus to name a few) on other topics (Marian dogmas, the Papacy, Prayer to and veneration of saints and angels to name a few). I'm not sure how many of these CA offers, but the "debate" with Akin (which CA has offered in the past) wasn't really a debate, but a discussion on the Bible Answer Man program. Still, it could be informative. If you're interested in getting some of the other debates that CA doesn't offer, you can PM me and I'll be happy to give you the website.