I've been getting into visual novels recently and am really enjoying them. My only gripe with the medium so far is that the games are very expensive and can go upwards to $80 or even $100 (not including high shipping costs from sites like J-list). Admittedly they're pretty long (haven't played one that's less than 30 hours), but I still feel the price is kind of heavy for something that's essentially a manga with sounds and re-used visuals.
The one I've enjoyed the most is Ever 17: The Out of Infinity. It's kind of like The Poseidon Adventure, but in a sci-fi setting and with a plot that has enough twists to even make M.Night Shyamalan raise an eyebrow. As the title suggests, the translation is pretty mediocre at times. But if you can get past that, you'll find arguably one of the best developed games of all time, let alone visual novel.
As you said, the medium is a bit dominated by romance at the moment. So if you absolutely can't stand it, then I'd suggest moving on since I can't think of any that don't have at least a little bit of romance in them, even Ever 17. That said, I strongly suggest you don't dismiss any of these as "dating simulators" just because they have romance or sexual content, more often than not these can end up being very small aspects of the game and are not the selling point by any means. Regardless, I'll mark the ones with sexual content with a heart icon.
Steins;Gate: If you haven't seen the anime, you should definitely check the visual novel it's based on. The premise is that a young unemployed scientist discovers a way to make his microwave send text messages to the past. Things take a turn for the worst when messages start changing the timeline and an evil organization tries to get the device for themselves. I know you're probably thinking "Oh god, not another time travel plot", but trust me, this story is anything but generic and is one of the better time travel-based stories out there. The art style is also very nice, albeit not as good as the anime.
: I haven't read this one myself, but I hear it's one of the best entry-level visual novels out there. All I know is that it has lots of action and a pretty good plot, here's a wikipedia link in case you want to find out more You are not allowed to view links. Register
It does have a lot of sexual content though.
When They Cry/Hirugashi: If you can get past the bad art style (and I do mean You are not allowed to view links. Register
to view.) and are into horror, then this is the one for you. The idea is that are someone who's just moved into what seems like a very peaceful town, when suddenly someone is murdered. You then witness the horrible deconstruction of everyone's minds as they slowly become more and more paranoid while trying to find out who the murderer is. There will be blood, and lots of it.
(just in case you do decide to try one out)
: As you may have heard about with all the controversy surrounding its release recently, Katawa is a pretty good game. The controversy is due to its premise, in that you play as a student whom suffers from a heart condition and has to go to a hospital. After you regain your health and leave the hospital, you are then sent to a "special" school where you will meet several love interests, all of which have their own disabilities and stories to associate them with. It's short, it's more mature than you'd think, and best of all: it's free.
CLANNAD: You should play this even if you already saw the anime, there are a lot of character arcs which weren't adapted to the anime so you're still missing about 40% of the story if you haven't played the vn. What makes CLANNAD great is that it asks you "what happens after high school?", which may not sound like much, but it's the only anime-verse story I can think of that goes beyond the school part of a relationship and goes on to show the whole story of a couple's life together. The other character arcs are extremely good as well and offer a decent amount of closure, even though most of them don't go past the school aspect of a relationship.
Also, here's a neat little website that lists all the best visual novels from greatest to least, as rated by the community You are not allowed to view links. Register
to view. There's a "no sexual content" filter on there as well.
PS: There's almost always more than one story/route in a visual novel, so make sure you set enough time to play a game multiple times through if you want to get the whole package.