Thanks for the awesome replies guys! I would like some clarifications on some points made in your answers;
1. I understand that the GND Filter would be useful for Landscape Photography, which I like to do sometime (20% of my photography maybe?) Would the investment be worth it or I should stick with Photomatix/HDR?
The GND filter seems to be an easy and inexpensive solution in theory. In practice it's another story. A cheap screw-in GND filter will come with a fixed position for the horizon -- when the time comes to actually use the filter that fixed horizon position will be wrong for your composition (Murphy's law). That pushes you toward a filter system like Singh and it's no longer inexpensive. Now you can position the horizon but when it comes time to use it there will be something raising up above the horizon that will be partially in and out of the ND zone and it'll show in your photo. Great in theory; poor in practice.
2. From the link posted about the HDR (Beginner Guide) Photomatix is said to be the best. Which one? Pro, Essential, Plug-In for LR only?
3. AEB has been suggested. I understand that I'll have to play with it to make my idea of it but generally speaker, it is a good method of should I always carry my Tripod and do it manually? Thing is, I don't have that problem everywhere, so the Tripod would be unwanted weight in some situation.
Photomatix is the most popular product, but there are many others available in a wide range of prices. As you've noted here HDR implies a tripod. The various software programs that do the blending are trying real hard and with some success to deal with slight registration errors between the multiple frames, but there's a bottom line here and it's a tripod. If you're really going to invest in the HDR route then start planning to lug the tripod everywhere you go.
4. Like stated in my first post, I pretty much alwas shoot in Aperture Priority. Someone suggested to "expose in the middle". Would that mean switching to Manual or the Exposure Compensation is enough? (Again, generally speaking as it is new stuff I'll have to test.)
Aperture priority or any other shooting mode that fundamentally relies on the camera's internal metering system is fine. Using EC to modify that exposure is likewise fine.
The problem you're having has been correctly identified by others here as a lighting contrast or dynamic range problem, hence the HDR recommendation -- the sample photo you posted was backlit. Beyond the two options noted above you can:
1. use flash with small and close subjects like a portrait.
2. walk away and use your time profitably to photograph something that is well lit. (I like this one).
3. capture a raw file and learn to expose and process it for maximum DR.
That final option doesn't require buying filters or special HDR software. You do need a good raw converter (you mentioned LR) and you need a supplemental image editor (like Photoshop) that can manage sophisticated masking. Using this option you want to get as much exposure on the sensor as possible without clipping the highlights. Runnah noted exposing for the sky and lifting the shadows and Kolia noted that you have max sensor data recorded at the highlight end. This suggests an exposure methodology known as ETTR (expose to the right) which references a histogram and the highlight or right side of the graph. In your original post you noted the noise you get from trying to lift the shadow detail too far. You're beginning with a very high contrast scene and to get maximum benefit from the camera sensor you want to make maximum efficient use of the sensor's recording ability.
Here's an example:
This photo is backlit with the light coming from the left and toward the camera. On the left you see the JPEG as delivered by the camera's image processor. The highlights are blown and if I had activated the camera's "blinkies" when I took the photo they would have been blinking. The camera histogram would have indicated the clipped highlights that you see in the sky near the horizon. This is an important point: The histogram you see in the camera references the camera's JPEG processing of the sensor capture and is not a fair indication of the actual data the sensor is recording. (Got to test your hardware). On the right is the photo I processed from the sensor raw capture using PhotoNinja and Photoshop. The camera sensor did in fact record full detail in the highlights which I was able to manipulate.
Using ETTR and raw captures can go along way to making HDR methods and GND filters unnecessary but it requires time, effort and skill. I still prefer option #2.
Joe