I'm probably as inexperienced as you when it comes to debate, so bear in mind that this is just a 2c rather than expertise here.

1) Try not to make them defensive
If their ego/self esteem is tied to them being right, this is going to make them less open to admitting to themselves that they are wrong. It will also direct more of their efforts to defending their ego rather than determining truth.

One way to do this might be to phrase things like "I personally can't accept that because I feel..." as it changes the focus of the argument. Rather than feel they have to save their face by throwing out rebuttals and make themselves look justified, you have instead raised their ego a challenge along the lines of "Here's my reason for disagreement and if you can understand this where I am coming from
It kind of gives them a positive encouragement to 'chase' where you're coming from and take your points on board in order to find an answer to them.

As it happens, I don't often take this approach in real life but have found it very effective when people have done it to me, so I should probably make a mental note to try it sometimes. Not that we should always do away with direct debate, but if in a particular debate ego appears to be getting in the way of understanding, a less direct approach can remedy this.

2) Don't expect instant results
Questions that we debate about tend to be incredibly complex with millions of angles. Even if you had the most perfect, ideal and winning argument, there's a good chance that are unlikely to fully comprehend it on the spot. Sometimes we are on auto-pilot arguing against what we expect to hear rather than genuinely taking the points on board. That doesn't mean that the arguments completely pass us by - there's a good chance that they have been absorbed into a subconscious which is likely to be bearing them in mind.
What I'm saying here is that the best of arguments take a while to sink in, but they gradually do and the listener finally finds that the point stands.

A good example is Paul Montana.
The guy was pre-suppositional theist.
He used to get into heated arguments with people in this community, both throwing their points back and forth, both sides feeling like that they were screaming at a brick wall. Over and over the same debates were repeated, seeming to get nowhere.

However, the points slowly sinking in.
Not so long ago Paul announced a rejection of pre-suppositionalist arguments for the same reasons that had been argued against him many times, declaring that belief in God was a matter of faith rather than logically proved through the arguments of the pre-suppositionalists.

So my advice would be not to worry if the argument gets heated or aggressive, so long as you can both cool off at the end of it and not mind that you still disagree. The points that were made likely had a subtle effect, sowed seeds in your opponents mind, that you can now leave to grow in their own time. You, likewise, will gradually reflect on the debate yourself. Even if you don't find their arguments convincing you will still be able to reflect on where they were coming from, why they didn't accept your arguments, and how to improve and explain it better the next time round.

Have you ever been in a situation where somebody made a suggestion or expressed skepticism in the form of a question?  If it was done well, expressed as a sincere request for information, did it seem not at all condescending and make you more inclined to respond thoughtfully?