Okay, there are a gazillion cameras you could learn on (new or used). There will be no one best option for you but a lot of good options. Also, I'm a long-time Nikon shooter so that's what i'll talk about b/c I don't know Canon or Sony as well.
1. You want to shoot landscapes. Than it's going to be much more than just the camera. Ideally you'll want a wide angle lens. And you'll want a tripod. You can get cheap tripods for $50--don't waste your money. You want a tripod you can put a DSLR on for 2 seconds and it will be rock-solid stable. A $40-$70 tripod is designed for videocameras so you can pan smoothly, not for a slow exposure. And that's what you're talking about with things like...star trails, blurred waterfalls, early sunrise/sunset and "blue hour" (dusk) photos.
Here is the equipment that I think you'd need to do most landscape photography: DSLR, tripod, polarizing filter, wide angle lens. Unless you buy them used, that's going to be tough to do for $350. Do NOT waste your money buying a cheap tripod--that's worse than no tripod at all.
2. You can actually get some decent used cameras and I suggest you look at those options. For instance, if you could pick up a used D70 or D90, they'd each do a terrific job for landscapes and they're good cameras to learn on. Here are the negatives: D70 is terrible at low light (compared to more updated cameras) and it's big. Pixel count isn't as good as the D90 or more recent cameras so it wouldn't be as good for poster-type photos. But you could get a good lens, a used D70 in good shape and a okay tripod (you'd probably pay the most for the tripod) for $350. A D90 gives you a higher megapixel body, better range for low light and a reputation as a camera body that just keeping on clicking forever and ever and ever. I have yet to run in to someone who had a D90 who found it to be a lemon. I know a professional wedding photographer who's backup body/second body is a D90. Both the D70 and D90 are big clunky bodies.
3. New cameras--I'd start with the D3100. Small body (b/c the autofocus motor is in the lens, not the body so if you're going to use AF you need a lens with an AF motor in it). More megapixels than a D70 or a D90. Smaller body so it's easier to carry around. Solid camera to learn on. It will probably cost you $350 with a kit lens that will work for landscape photography.
4. You'd talked about hearing some kit lens weren't sharp. Here's the deal: all kit lens are compromises. They're usually zoom lens and none are pro quality. For a budget of $350, you can't buy even just a professional lens. And that's okay b/c for landscape photography, you can afford to put your camera on a tripod, use a slower exposure and not require an f2.8 200mm zoom. Sharpest lens will usually be prime lens but they almost never come with kits. And the cheapest wide angle Nikon lens I know of (35mm f1.8) will cost you $175 new (it's a really good bargain though). So you're probably going to be shooting with a kit lens initially.