Today I read the most excellent interview between a Muslim and a Christian that I have ever read. It exemplified the genuinely respectful engagement with other religions and our friends who adhere to those religions which I aspire to in my own research of other religions and irreligious philosophies.
I have sought to engage with many other religious traditions respectfully, particularly Buddhism. I am also currently researching Hinduism and Islam more deeply than simply reading their texts, and I have written a great deal about atheism, both contemporary and otherwise. I have even written about Satanism in a way that is respectful and uses their own sources to defend Satanists from an accusation made by their detractors.
From my own successes and failures in the area of inter-religious dialogue, I have learned a great deal about what it is to respect another religion as one learns about it. I have also learned a great deal about what it is to disrespect another religion. There are obvious and subtle ways to disrespect another religion; one of the most common is to spread misinformation about it, and this could be either obvious or subtle, but it is usually not so obviously disrespectful as insulting a religion.
The spreading of misinformation about a religion is generally not malicious. I would go so far as to say that it's more likely that the people who spread misinformation about religions are acting in good faith. I've accidentally done it myself, and when I learn that my previous understanding of a religion is incorrect, overly simplistic, or not quite as evidence-based as it ought to be, I renew my commitment to being more respectful of that religion in the future by sharing more accurate information about it when the opportunity arises.
The antidote to this disrespectful spreading of misinformation is research. Don't take your smart-ass friend's word for it that the Buddha was an atheist or that Catholics worship Mary or that the Quran is more full of violence than the Bible. About two minutes of research on Wikipedia can clear that sort of misinformation right up. But if you want to be even more respectful than just not spreading misinformation, then more than just fact-checking yourself with readily available general sources will be necessary.
If the founding texts of a religious tradition were not written in English originally (and the vast majority were not), then find a scholarly translation of the texts into English by someone who is committed to preserving the values of that religious tradition. Or if you happen to read Sanskrit or Greek, then just read the originals if we still have them. And beyond that, read commentaries on the texts by practitioners of the religion who have been practicing their religion for most of their lives.
But wait, there's more! In order to more fully understand a religion, it's important to experience it. My visit to a Vajrayana Buddhist monastery, my praying with Muslim friends in their traditional form of prayer, and my monthly attendance at an Antiochian Orthodox Divine Liturgy was educational in a way that simply reading the texts and great scholars of those religious traditions could not effect. It is necessary to consider the question of whether certain religious practices might harm you, of course.
I don't recommend engaging in religious practices which you have good reason to think might be harmful. I also strongly recommend that you do not try to engage in religious practices which longtime practitioners of the religion do not think you are ready for. Such a thing ventures too far in the other direction and also becomes disrespectful. And speaking of becoming disrespectful by trying too hard...
One of the most common ways to disrespect another religion these days is by trying to compliment it. Calling Islam "a religion of peace" may seem like a great way to counter anti-Muslim sentiment, but it is inevitably reductive; it puts Islam in a box and bids it stay there for its own good. Islam has a rich tradition with many branches and elements, and there are many ways of understanding sharia with regard to what sorts of violence are halal and how Muslims ought to achieve peace. It is unfair to the Prophet and those who heed the message of the Quran to lessen Islam in order to present it as something palatable to everyone.
But sometimes other religions don't even get a compliment. Recently, I was presented with the claim that Jesus, Muhammed, and the Buddha all taught the same message of love, with the implication that these religions are all the same thing and that we need to look past the divisiveness of sectarians who believe that these religions are actually different. The people who advance this claim are generally sincerely trying to be respectful to these religions. They fall into the trap of spreading misinformation about a religion in order to promote unity.
It's disrespectful to any religion to reduce it (as many in the West do) to a slightly different version of the same religion you grew up in and later abandoned. Islam has its own set of truth claims, some of which (in the Quran, specifically the Sura on Mary) directly contradict Christian teachings, and textual analysis shows that it spends a lot less time talking about love than Christian writings. Also, the Buddha taught an incredibly robust path to the liberation of mind from the cycle of death and rebirth that ought not to be waved away as just another message of love.
It is far more respectful to politely state your disagreement, to take the risks associated with speaking the truth with love, to regard a religion as worthy of the time and trouble it takes to make a reasoned argument for not following it. Do the religions you don't follow the kindness of treating them as what William James called a "live option" which you at least consider carefully before rejecting and which you are willing to re-examine when new evidence comes to light. It is far more respectful to other religions to treat them as you would want your own religion or philosophy to be treated: with honesty and with an effort to understand your views fully.
My advice is to respect other religions enough to research them, to take their truth claims seriously and examine the evidence for them, and to disagree with them honestly where you cannot agree with their truth claims. The greatest respect we can pay to another religion is not remaking it in the image of our own religion and not reducing it to merely a lesser version of our religion or philosophy, but rather to take as it is, accepting it in the fullness of the unique beauty of its practices and claims to truth before evaluating it.