To get started, I read Mindfullness in Plain English , which is a pretty practical, no-nonsense guide with minimal New Age woo. It addresses a lot of the problems and questions a beginner will have getting started.
I can't really claim any long-lived effects or benefits of it. As I said, I'm very skeptical by nature, so I tend to be wary of attributing anything to it. One thing that's very clear and noticeable though is that I feel very mentally different during and afterward. My mind is quiet in a way that it otherwise never is, and I feel like I perceive the world differently somehow (maybe more clearly or more at face value.) It reminds a lot of a hyper-focused flow state, except it's non-directional. It's like I'm focused on everything and nothing all at once.
After I started, I did some reading on wikipedia and found that meditation has been found in scientific studies to have certain benefits and neurological effects. It's apparently even a treatment doctors recommend for handling stress and anxiety. That has helped me stick with it, because the meditation community is full of New Age bullshit. It's nice to know that there's more to it than that.
The effects of meditation are best compared to those of sleep. A good night's sleep doesn't make up for lousy sleeping habits the rest of the week. But getting enough sleep _regularly_ is key to being healthy, happy, and productive.
Ditto meditation. Trying it once in a while is likely as waste of time. But practicing it regularly (and getting better at it, which you will, with practice) can have positive effects, over time.
As an aside - Zen takes a good deal of commitment, but it's the least 'bullshity' of all the approaches to meditation/mindfulness I've encountered. Eight Gates of Zen is a great intro: http://www.mro.org/zmm/training/eightgates.php
In the Eastern religions this is very much what is referred to as "faith"; the study and analysis of meditation in this manner is intellectual, but meditation itself if very much anti-intellectual and seeks for the non attachment of intellectual concepts (a difficult concept to grasp for those of us raised in the rationalist philosophies of the West).
It doesn't mean that you can't measure it; it means that, aside from the sensations which come to you of well-being and effects that come to you from direct observation, you shouldn't be using it as your primary guide to the effectivity of meditation. Fortunately there's 3000 years of incredibly descriptive and academic Buddhist and Hinduist texts which talk about this in amazing depth, and have zero usefulness for your personal progress.
Right now, whatever comes up my mind i just let it be and think nothing of it. Null.
Oh, by the way, I also do self-hypnosis at the end of meditation.
When I think about it, it's hard to believe that improving one's state of mind and mood wouldn't have a number of benefits.
However, I will point this out. The point is to be. To experience all of reality: to stop avoiding the things you don't want to experience, and to let go of the things you want to go on forever.
When you say, "the religious aspects are a turnoff", those are fantastic shadows to investigate. And I don't mean, investigate them as in some sort of Socratic dialogue with yourself. I mean getting in touch with the feeling of being "turned off", locating the physical sensation that give rise to your mind's interpretation as the emotion of "turned off", and really watching it with your mind without blinking. These uncomfortable feelings are where you'll find the most intense insights about yourself. And it doesn't matter whether you yourself believe the religion, or whether the religious aspects are an illusion or not, because the feeling of being "turned off" is real for you. That's all that's needed to start.
This is a shame. Within the frivolous details we debate so much religion holds a wealth of wisdom and value that anyone can apply to their lives and benefit from.
I took a similar stance when getting into meditation. I was looking for a way to deal with stress and I was certainly not "religious". I quickly found that modern/western Buddhism is stripped free of a lot of the Asian cultural traditions that I perceived as religion.
I have to agree. Many of the fruits of the Enlightenment have been very positive, and the oppressive and totalitarian aspects of what passes for organized religion deserve to be rejected in the strongest possible terms.
However, the viciously anti-religious attitude that the Enlightenment has spawned and which has had a resurgence recently thanks to narrow-minded bigots like Dawkins, Harris, and Hitchens, the mainstream media's relentless focus on church scandals, some forms of religious extremism and supposedly religiously inspired terrorism (all of which I condemn in no uncertain terms) has led a lot of people to reject all of religion wholesale -- which is throwing the baby out with the bath water.
There are many profound, thought provoking, inspirational, and beautiful ideas, images, and messages in religion. And there's an incredibly wide and varied range of different types of religion -- to an extent that most people are simply not aware of.
Despite what we may see on the news or read in books by the likes of Dawkins, Hitchens, or Harris, religion is not all about domination, pedophilia, or unquestioning obedience to some leader or dogma.
Unfortunately, many people do buy in to such myopic visions of religion, mostly through ignorance and propaganda. As a result, they're really missing out on the parts of religion which are positive, constructive, profound, and even liberating.
That said, I'm not a believer myself. I'm agnostic, but a student of the history, techniques, and beliefs of religion (among many other things). I want to learn about every aspect of religion -- good and bad, light and dark. Few fields as large as religion are as simple as they might first appear.
 - https://en.wikipedia.org/wiki/The_enlightenment
This is a shame. Within the frivolous details we debate
so much religion holds a wealth of wisdom and value
that anyone can apply to their lives and benefit from.
I agree: there's a lot of wisdom to be learned from religions.