Normally you would use sRGB mode. It is the most common denominator. Keep in mind that this mode is not calibrated, so your sRGB colors will be different from other sRGB colors. They should be closer.
Once in sRGB mode your monitor may not be able to show colors which are outside of sRGB color-space which is why sRGB is not the default mode. The truly odd thing is that those particular colors are kind of random until you calibrate your display!
If you do calibrate your display, you can revert back to the default mode (called Native on some monitors) and then all color-management-aware application will be able to:
- Show colors outside of sRGB with good accuracy.
- Show sRGB colors with good accuracy.
However, non color-managed applications will still show colors wrong and they will be MORE wrong in default mode than in sRGB mode. So which one to use in the end will depend on what non color-managed applications you use to look at images. This may include your web-browser depending on which one and version you use.