Your problem is caused by using an exposure that is too long for the type of subject material you are photographing. As the name indicates, the "scenic" or landscape mode you are using implies subjects that are distant and not moving. And, given the long exposure, you are correct in noting that a tripod results in the best image.
The quickest fix would be to place your camera in "Auto" mode and let the camera decide the best settings for your picture. With this method you will find that you will obtain adequate results MOST of the time. For the best results, changing your camera's settings manually will result in the best pictures. You can also try some of the other pre-programmed modes depending on your subject matter. For instance, the sports setting would be good for any picture that contained objects in motion or needed to be taken in low light conditions. What the pre-programmed modes do is adjust some of the settings I will describe below.
A little lesson is in order. Your digital camera borrows some conventions from their film-based cousins. Film speed, for instance, or AS settings refer in film photography to the sensitivity of the film to light. The higher the AS number, the more sensitive the film is and, therefore, the shorter amount of time the shutter needs to be open to expose the film. In the "scenic" mode you used, the AS setting was probably 100 or 200.
To photograph moving objects, or scenes with lower ambient light settings, you would want to manually adjust your AS settings to 400 or more depending on your camera. I believe yours has a setting of 800. You will find that these higher speed settings will get rid of the "shakes" that you refer to and the motion trails -- but it does come at a cost. The higher sensitivity will result in images that tend to look "flatter" with less color saturation and will often be "grainy".
The good news is that since your photographs are digital, you can process the images after the fact and fix many of these problems. There are a number of great plug-ins and programs available to help fix digital images. I can personally recommend the plug-ins available from Kodak. There are two versions, a Professional and standard flavor. The GEM filter is specifically designed to remove digital "noise" from your images.
Another convention borrowed from film cameras is the concept of F-Stops. Like the AS settings, the F-Stop affects the amount of light the film is exposed to. In film photography the F-stop refers to how wide the shutter opens. The bigger the F-Stop (higher the number) the wider the shutter opens and the greater the amount of light over a fixed amount of time (length of exposure).
Typically, in outdoor/motion photography you would adjust the film speed (AS) and not the F-Stop. But in a more controlled setting (portraits/still-life), you could open the F-Stop and still keep a low film speed to compensate for low light and still not have to resort to extraordinarily long exposure times.
So, what's the bottom line? Digital film is free to develop... so experiment... experiment... experiment. Many editing packages will report the settings used in your images (AS, F-Stop, Exposure). Use your camera's pre-programmed settings and then examine the settings the camera chose and why. Also, I recommend buying several higher capacity memory cards. If you can afford it, the added capacity will allow you to shoot in RAW mode. The advantage is that RAW mode records exactly what the lens sees (regardless of what AS number or F-Stop is selected). You can then experiment after the image is taken by adjusting exposure times, F-Stops and film speed. Have fun.