Can I create HDR images with a Sony DSC-H10, and how do I do it?

We'll leave your second sentence for another debate :) but I highly recommend you at least get a photography book (or an introductory class) which will teach you what to do to get great photos with almost any camera.

To get an HDR image:

1. Set your camera on a tripod.
2. Use either Manual (M) or Program (P) exposure modes.
3. It is highly recommended you enable the 2s self-timer to avoid shake.
4. Fix the ISO to a low value, like 100, to maximize image quality.
5. Repeatedly take a few shots by varying the exposure as follows:
   1. For P mode, take 3 shots one with EC at -2, the other at 0 and the last at +2. This will work for scenes which exceed the DR of your camera but a little. OR
   2. For M mode, you can capture way more dynamic range using more images. In this case, just keep changing your shutter-speed and take a shot every 3 or 4 stops. For example: 1/2000s, 1/250s, 1/30s, 1/4s, up to the limit of your camera. y or Daylight for example)

If you are shooting a very dark scene or something with deep shadows, you may need to increase the ISO on your last shots.

EDIT Missing the final steps:

1. Take the set of images shot and merge them into an HDR image. There are a number of software to do this and, including Photoshop, in case you already have it. Otherwise, just Google for HDR software.
2. The final step is to tone-map the HDR image. This converts it into a Low Dynamic-Range images which can be shown on computer screens and printed.
3. Alternately, you can perform something similar in one step using Exposure Fusion.

NOTE: On most cameras you would use A (Aperture-Priority) mode instead of P (Program) mode for automatic exposure. However, considering your camera does not have a physical aperture, it makes no difference and is actually the reason why there is no A mode on your camera.