In order to preserve maximum details and future editabiliy, it is best to save your images in RAW format, if your camera supports that. That said, it is generally far more demanding in terms of memory per image requirements, hence can limit your max number of images.

For JPEGs, there are generally two factors that determine the image quality (IQ). The first one is resolution. Obviously, the higher the resolution, the more details can be captured in your image. That said, you mentioned uncropped 8x10 prints max. A standard lab printer prints at around 300dpi. This translates to 2400x3000 pixels, or ~8 Mpix. If this is your max output format, then you can reduce the capture resolution to around that number (*).

The second factor is the quality, or compression, which is determined by the lossy part of the JPEG compressor. The compressor, generally, is throwing away the information of the higher frequencies in the image. Considering this, you can judge by your scene - if it is a scene comprised of tiny/fine details (e.g., a tree/grass, head-hair, etc.) you may want to keep quality at superfine.

Otherwise (shooting you car, for example, or other relatively smooth objects) you can lower the quality to save space.

(*) Note that when doing post-processing work that includes rotation or other pixel-destructive actions, having more pixels to work with will give you better final output.

Update in response to OP's commet:

Well, there are a few factors here. When dealing with this matter, I assume the consideration of technologies of similar age. A new sensor will most probably outperform an old sensor with a higher resolution. As the sensels shrink, their noise sensitivity increases up to the point that image details are lost in the sea of noise at the pixel level*. Thus, from some point increasing the resolution becomes meaningless. But, newer sensors have better noise immunity at the pixel level.

Different brands and models use some variations on sensor technology. The types I am aware of are standard CMOS, CCD and backside-illuminated CMOS. CCD and CMOS are comparable in noise performance these days. BSI is considered a newer technology which increases the amount of light gathered by the sensel, hence its noise immunity. In summary, there may be a difference in actual resolution between models but for same technology, it will increase your captured resolution (see next paragraph).

Remember that a sensor is not just the silicon piece but also a stack of filters and microlenses. One of the filters is a low-pass (anti-aliasing) filter that cuts the image's high frequencies before it reaches the sensels. This alone will suggest that the optical/analog image itself is more detailed in a higher res sensor.

Another point that affects actual resolution is the resolving power of the lens. Nowadays, most models have sensors that have outreached the resolving power of most lenses (absolutely definitely when you talk about high MP compacts or smartphones). This means that the benefit in increasing res is marginal, but it is there. @Matt Grum explained in on of his posts (I'll try to find it later) that the captured image is the convolution of the lens image (signal) and the sensor sampling function. As such, there will always be some improvement with increasing res, but it is questionable if you can take advantage on that.

As for the subject being captured - obviously (really, this time) if your subject has no details, then I don't see how increasing the resolution will improve the final image (digital interpolation will work just as well). I touched this point in the first part of the answer.

To sum-up: technology, as applied to different models even among a single manufacturer's line, can affect the resolving power of the sensor. When comparing same technology, one can show that increasing resolution does increase the amount of details up to the noise floor of the sensel. How detailed your subject is will definitely affect how detailed your image is.

Resolution alone is not the only player in the game, and when choosing a camera one needs to consider all the other parameters (lens, filters, processor, etc). Your question intent seems to be the choice of settings in the context of a given camera, which is what my original answer addressed.

Update II: Here's Matt's answer: Do megapixels matter with modern sensor technology?