To me the best way to win a moral argument is not by verbal exchange, it's by actions. One can scream relentlessly that they the most moral person known to man, but their actions may tell otherwise.

Found this normally works.

to be a convincing moral person, known by your acts, not words.

Name-calling, rock throwing, and when all else fails, an aluminum baseball bat.

Jackboots and rifles seem effective.

First you must assess if the conversation will be civil, respectful of everyones opinion and time is not rushed. Talking in turns, for example, will help here.
Second, a clear value system must be put in place to weigh the arguments i.e. reducing human suffering/life, environmental impact, future generations, freedom of expression, human rights, etc...
lastly, lots of alcohol to loosen inhibition (wink!)

When I get some free time (Wednesday) I'll put together a plan for you. It's fairly easy to follow and then all you need to do is be sceptical. Always get the other side to ask questions about their own thoughts.

Sam Harris is working on an evidence based system of morals and/or ethics. IMO that's an undertaking for several lifetimes, but a worthwhile one.

Don't like to argue morality or religion.

I think the key word is argument. As the quote says, it is better to be silent and be thought a fool than to open your mouth and confirm it. Most people who want to argue their perceptions/morality are doing it more so to convince themselves than someone else. There are some very useful counseling techniques that work very well with people who want to argue, such as letting them “run the tape out”, parroting, summarizing, and active listening. Let them keep talking until they exhaust themselves and they usually say something foolish to get you to engage.

Love this quote by George Bernard Shaw:

I learned long ago not to wrestle with a pig. You get dirty, and besides, the pig likes it.

Be true to yourself and live your morals.

Argue on the basis of hurting or not hurting people. If they try to drag it back to religion, just say you don't want to hear what they think god says, but what they have to say. Sometimes, it's a struggle to keep the conversation on track. (I'm in a closed argument group on fb, and some people think that name slinging is a good way to argue, and another just give the name slingers a hug meme. They catch on fast.)

Slowly....and with an open mind.

Listen and respect

I'd say math. Do the math and see which option inconveniences the fewest people, or (what's the antonym of inconvenience?) conveniences the most, degrees need to be considered ofcourse. There's your moral compass.

Accept morality as Subjective, argue within that scope.
Use logic and win the day.

Not hard.

I don't think you can have a convincing moral argument (as in persuading someone to your point of view). I do think one can share his/her point of view which could open someone up to a different perspective.

be passionate about what youre saying?

Logic and honor is the only side one can be, to be on the right side.

Try not to be personal and insulting .. if that fails make sure you have a bigger gun .!!!!!

Evidence

I watched a video on arguments on the Philosophy Crash Course yesterday. It was good. You need empirical evidence for your premise and conclusion.

Do your research befor embarking on this one, else you might end up being embarresed by not having answers.