You will be able to get a fine camera for far under that budget. And from what I gather you are looking for a point and shoot camera. You are right in trying to adjust the ISO high like 1600 or so for the night photos, the only caveat is that with a high ISO you get pictures that are more grainy. But I think I can give you a few recommendations.
A few tips that are general for any digital point and shoot.
- Anything above 7-8 megapixels is fine. Dont buy a camera just for the megapixel max, more megapixels take up more memory on your card which is fine since you can just delete what you dont want, but dont let it be a deciding factor. Anything above 7 is going to give you fine 4x6 or 8x10 printouts. If you are printing things larger you can step up but most people cant tell the difference.
- Very Important buy a camera that has Image Stabilization. Canon calls it IS, Sony calls it steady shot, every company has a name for it but basically it helps to reduce the shakiness of the photographer. Make sure it is optical not digital image stabilization on whatever you buy.
- When taking pictures use two hands when you can, it helps to stabilize the camera, if you are at a game where the action is down in the field a tripod will be a great help to you.
- Press the shutter button (the button you use to take the pictures) halfway and the camera usually makes a noise. Then continue to press it all the way to take the picture. This is called focus lock. For example go ahead and find what you are going to take a picture of and then try to press the button half way, when you are ready to take a picture press it all the way. This involves some anticipation on your part but it may help you take better pictures.
- Night shots with a point and shoot are difficult. The reason is that for a good night shot everything needs to be still and the light has to be just right. The camera keeps its shutter open and tries to gather as much light as possible before recording the picture due to the fact that at night there is low light. Since the shutter is open for 1 second for example instead of a fraction of a second movement causes a picture to be blurry and weird. Flashes at night are really just terrible in my opinion, plus a flash on a point and shoot is probably too weak. Additionally if the ISO is set real high as noted above it produces a grainy picture.
- Action shots are hit and miss as well. The auto settings do the best hey can but the thing to remember for action shots is that you have to be ready to take them, keep the camera on the subject preferably focus locked so that when you need to take the shot you only have to press the button half way to complete the shot. The auto setting tries to help the user by either taking multiple shots in succession to try to get the action shot or by increasing the shutter speed so it takes the picture within a fraction of a section to capture the action, this works better when the scene is well lit like an outside sporting event. This way you can keep the iso low like 200-400 to keep the picture crisp
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- The Canon PowerShot SD780IS is an excellent camera that my wife uses. It is point and shoot, very simple, and very small. It can be used one handed without much trouble and without much shake since it is light. It also comes in several colors which my wife loved. On a side note about size I found out a long time ago if a camera is too large we wont usually carry it and we wont have it for the times we get candid shots. This camera is usually priced around 200-250 dollars. Amazon has the silver one for 207.32 now. The downside to this camera is that since it is small the flash is only going to be good at a short range, mostly close indoor shots. The flash is useless outside. The 3x zoom is sufficient for us but you may not get the real close shots you want to a football field from the stands.
- The Canon SD 980 IS is similar to the above but with these major differences: it is larger, the zoom is longer about x5, and the lens is wide-angle. This wide angle is great if you are taking a scene or you want to capture the whole field. Imagine taking a picture of a room and looking at the left and right edge of the photo and remember what is on the edge of the picture. Then imagine taking a few steps back and retaking the picture, you get more into the shot. With the wide angle lens you are the same distance from the subject but you capture more field. About $300 on Amazon.com.
- The PowerShot SX120 IS is what you are going to want for great zoom. Essentially this is a larger camera than the previous one I suggested. It still has IS and it has 10 megapixels. The plus about this camera is that you will getter a slightly better flash, for someone with larger hands this may also be easier to handle. As far as I know it only comes in black. The big advantage is the 10x zoom. This one is about 250 on amazon.com
Obviously I have recommended two canon cameras. Other brands I currently use are sony, olympus, nikon, and in the past fujifilm and ricoh. Personally for all around ease of use, price, and functionality I suggest canons. I think that none of the point and shoot cameras regardless of brand are going to give you great night or action shots. This is something you will most likely need to play around with. I highly suggest reading the manual. Another option I have used for action shots is I video tape the action and then use my computer to get the action shot since I have it recorded as video. Still not a great option for action though. The best option for action is a larger Digital SLR camera, but I think that this does not really fit what you are looking for in a camera.
Also I tried to pick cameras that were about 300 or less from amazon.com which I picked as a reference because it is pretty consistent in price. You will need to shop around for a good size memory card as well. Look for the largest card you can find, probably going to be around 8gb so estimate about $50-$100. This way you can just keep shooting pictures and not have to worry about running out of space. Within your less than $500 budget you can also add a good camera case, and perhaps a tripod, or a gorilla pod (a flexible tripod so you can mount the camera on just about anything).
Search websites such as techbargains.com, fatwallet.com, or slickdeals.net for help in getting the lowest price. Other people may suggest more expensive cameras but the following is how I feel about this practice. In general I don't usually buy the best or most expensive cameras I buy used, refurbished high end digital slr cameras sub $800, or I buy moderately priced point and shoot cameras sub $400. This way I can buy a new camera every few years if needed. I don't believe in spending a whole bunch of money and then buying a new camera in 5-10 years. Technology changes too quick and I would rather have something usable all the time instead of the best technology for just 1-2 years. Hope this has helped you in your search. Sorry for the lengthy answer.