Well, of course there are the Myst games - the classic series that sparked the whole genre. With the possible exception of the fifth game (which is more of a tacked-on epilogue to the series; the fourth one has a very satisfactory ending to the story), they're absolutely brilliant. Gorgeous graphics that are way ahead of their time period, really challenging puzzles, and a very compelling storyline with superb acting (for the most part - especially in Exile, where the villain is played by none other than Brad Dourif!). As for religious content, I don't think there's any worries there - the original creators are Mormon, so they have a really refreshing take on the creation of worlds. It comes out more in the supplementary novels, but you can still see this very biblical take on things in the games, especially in Riven. There are occasionally signs of indigenous religions and so forth, but I don't think there's anything too problematic.
Broken Sword: The Shadow of the Templars is an old game, but pretty good. I played the remake for the Wii, so I don't really know how the PC version differs, but the story was cool enough, and it has a very nostalgic feeling now. (The voice acting is hilarious, for one thing. My sisters and I like to quote the main character saying, "The door...was locked."
) It feels very much like you're playing an action movie. I don't think there's anything more worrisome than something you'd find in an Indiana Jones movie, for example.
Sanitarium would probably only appeal to you if you like horror games. It's another old one, but I found it fascinating and really enjoyable to play despite the graphics and voice acting (and the sheer difficulty of getting such an old game to work on a Windows 7). It follows a man who wakes up in an insane asylum with amnesia, and then really weird things start to happen. You could take some of the stuff, like the statue of an angel coming to life, as religious, but I took most of it as explorations of what was going on in the guy's broken mind. Again, it would probably only appeal to horror fans.
The Last Express is a phenomenal game - once again, an old classic.The gameplay can be a bit wonky, but it's a really intriguing mystery that you have to figure out, involving time travel and figuring out what happened by piecing together different people's perspectives on the events. It feels a lot like Murder on the Orient Express. This game is also really cool to play, because it uses a lot of rotoscoping. An outdated technique, sure, but still really cool to see. Oh, and this is one of the only games that actually progresses in real time
I'm not really sure if To the Moon counts as "point-and-click," since it's pixellated and mostly controls like an old JRPG, but it has no (real) battles and focuses primarily on the story and a few puzzles. But mostly the story. Anyway, To the Moon is an absolutely
gorgeous indie game set in the future, where people can live out their dreams in their mind as they're dying. So you get to jump into this old man, relive his life, and help him find peace and closure before the end. Everything about it is beautiful - the story, the graphics, the music....