There are plenty of pocket cameras that do what you describe. For example, my Canon S90 provides exposure modes like P (program mode, sets aperture and shutter), Av (aperture value, allows you to set aperture), Tv (time value, allow you to set shutter), M (manual, you set everything).
All pocket cameras will record the settings into the EXIF data in the file. Most should allow you to check the settings on the camera's display screen.
Pocket cameras are available in a wide range of prices and capabilities. You should narrow down your price range, lens type, etc. and then look for a model with specific features that you require.