Recently I bought my first DSLR camera. It is a beginner's one, a Sony Alpha A230 kit. To tell the truth, my shots are no more crisp than those I used to take with my old point and shoot Nikon Coolpix.
Is it possible to take really sharp photos without buying an expensive lens? What would be your advice for that? (I'm really trying to go beyond automatic mode, though with little success so far).
There are several factors to getting a sharp image, the lens is only one of them.
Lighting
If you have a well-lit shot it will be much easier to get the crispness you want because you can get higher contrast lines.
Shutter speed
This is actually pretty closely related to a well lit shot, but you want to make sure your shutter speed is fast enough to avoid camera shake, which will cause blurring. A good place to start is with a reciprocal of the lens length (i.e. for a 50mm lens, at least 1/50s).
Camera shake
If you can't get lighting good enough to use a fast shutter speed, then you just have to do your best to avoid camera shake. This means working on you technique, both with holding your camera, and with pressing the shutter, or if your lens has image stabilization (or vibration reduction) then use it. When none of the hand held methods work, then use a tripod.
Aperture
Every lens will have a aperture at which it is the sharpest, a good place to start is in the middle of the range, typically around f/8. This is a bit complicated, because there are 2 opposing elements at play, depth of field, and diffraction. Ken Rockwell has an in depth article about this if you really want to study it.
Lens Quality
Some lenses are just better quality and will have less aberrations, so a shot taken with a kit lens will probably never be as sharp as a correctly-exposed image from a Leica 50mm f/1.4. That is not to say that any shot taken with a superb lens will be better than any shot from a kit lens, because that is certainly false. It does mean that the upper range is higher of some lenses.
Use a tripod (or a sufficently fast shutter speed), use low ISO, stop down to around f/8 (many cheap lenses perform a lot better in this range), do some post processing to add sharpness, local contrast, pop, etc.
Find out which cheap lenses are good that you can find at pawn shops or eBay. My recommendation, vintage lenses: See here: Where can I find reviews of lenses?
How are you taking your shots? There are a variety of factors that affect the clarity of a shot that you might want to work with. Lighting, aperture, shutter speed, and the type of thing you are shooting all play into how your shots end up looking.
First off, lighting. If you are regularly shooting in dim lighting, and you have a slower lens (i.e. f/4 maximum aperture or tighter) you might really need to consider getting a better lens. Something that has a faster maximum aperture, such as f/2.8 or f/1.4. Wider apertures (smaller numbers) allow more light in, so night shooting or photographing indoor sports greatly benefit from wider apertures.
Second, you might look at what shutter speed you are using. A general rule of thumb is that a shutter speed at least as fast as one over the focal length of a lens necessary to take a "stable" shot. If you have an 18-70mm lens at 70mm, then you would need a minimum shutter speed of 1/70 to take a stable shot. In low light, that may be too fast to capture a bright enough shot with an aperture of f/5.6. On the other hand, at 18mm, you would only need a shutter speed of about 1/20 at f/3.5 to take a stable shot.
The higher the shutter speed at a particular lighting, the wider the aperture must be to keep a constant exposure. Eventually you will hit the limitations of your aperture, and you will need additional lighting to keep the shot bright enough.
If you are taking shots at night or indoors, try using your flash. Even in lit conditions like indoor sports, a flash can help brighten up a scene and freeze motion.
As a final resort, you can increase your sensors ISO setting. Automatic modes tend to change the ISO as needed to accommodate lighting conditions. Once you start using semi-automatic and manual modes, however, you may need to adjust ISO manually as well. Some cameras also limit the maximum ISO that may be selected automatically, which is usually about ISO 400, possibly ISO 800. If you need to use a higher ISO setting, you will likely need to choose it manually even in a semi-automatic camera mode.
A high ISO setting will allow you to use a faster shutter speed in lower light. Where you may only be able to use a shutter speed of 1/20th of a second at ISO 100, you could use a shutter speed of 1/160th of a second at ISO 800.
If you can't improve your shots by adjusting the aperture and shutter, or by using flash, you might want to get a tripod. Attaching your camera to a tripod greatly improves the stability of your shots over hand-held.
You might also try using a semi-automatic camera mode rather than full manual. Most cameras support aperture priority, shutter priority, and program modes. Aperture priority allows you to choose the aperture, and it will automatically set the shutter speed for you. This is probably the most common camera mode used by most amateurs and professionals alike. If you need control over motion, shutter priority is probably better, as it will allow you to choose a shutter speed and automatically set the aperture for you. Flash in combination with shutter speed will give you more control over freezing motion. Program mode is a mostly automatic mode that usually provides some control over exposure compensation. You can expose at zero compensation, or compensate +/- 1-2 stops (perhaps more), and the camera will automatically set shutter and aperture for you, accommodating your compensation.
Try a cheap lens! Cheap doesn't necessarily correspond to "bad."
The used market is full of older lenses that are compatible with modern DSLRs, and are often available quite cheaply. Sony, in particular, can use any Minolta AF-mount lens, including autofocus, and the 50mm f/1.7 is likely sharper than the typical kit zoom lens. At around $100 on eBay, I wouldn't call it expensive (at least not in the context of lenses). Research might reveal a few other good lenses compatible with your camera.
Buy a remote cable instead of clicking the camera. This remote shutter will ensure that there are no movements and this will definitely result it much sharper pictures.
Here's how I do it, handheld:
Make sure your shake reduction is on, for one. Sony has it in the body like Pentax and it helps. Without it, the general rule of thumb is a shutter speed no slower than the reciprocal of the lens length meaning that if the lens is 100mm, then you should have a shutter speed of at least 1/100th of a second otherwise you should be on a tripod. With SR, you can go a bit slower. However, make sure you press the shutter, not stab it.
Bear in mind, also with Sony, you can use Minolta lenses. You can find some very good deals on older Minolta lenses on eBay and Craigslist, saving a bundle without giving up quality. There's lots of old glass out there that is top notch, used isn't a bad way to go (I've bought some real gems for my Pentax this way).
My final suggestion would be to take rapid, multiple, shots of the subject if you can. Media has gotten pretty cheap, so don't be afraid to pump a few shots in as you increase your likelihood of get a crisp image.
Beyond that, I agree with the other responders.
What kinds of lighting situations are you shooting in? Are you using a tripod, or at least following best-practices to minimize camera shake while shooting hand-held?
If you're shooting with natural light and your camera is held steady, you should be getting sharp shots.
A different lens might help matters if you shoot low light and the lens has a wider aperture than what you have now. An expensive lens might have vibration reduction (VR) which should also help.
But be sure you have the cheap and simple basics covered first.
