Congratulations on getting a scope, they are a valuable tool that will go along way in your understanding of electronics.
In answer to your questions, first regarding grounding. The scope requires a reference (ground) to which the signal (probe) is measured against. The most common method of connecting this ground is to use the "small crocodile clamp". This method should be fine for basic measurements of slow signals. If you compare grounding this way to the one you used where you connected the ground to the input plug, you will notice that the loop of wire formed by the signal to ground connection is much smaller with the "small crocodile clamp". In a nutshell, smaller is better as the larger this loop area, the more noise you have contaminating your measurement.
Second, the sawtooth waveform you are seeing on the 9V supply is probaly the ripple voltage that "rides" on the DC output. Most supplies have a specification for the ripple, typically measured in milivolts (mV). To measure this with the scope, you need to determine what you have set as the "volts per division" on your scope. Then using the grid lines on the scope display, observe the signal. One block or division on the grid lines represents the X volts of your volts per division setting. For example, is you are measuring at 1 volt per division and the sawtooth is 1/2 of a block tall, then you have a 1/2 volt peak - peak sawtooth.
Third, regarding seeing the noise from a switch, this is where the time base (or seconds per division) comes into play. The time base settings determines how much time each grid block on the display represents (measured from left to right). The switch bounce is probably on the order of a few miliseconds. What you can experiment with is setting a trigger point, which is a voltage at which the scope will "fire" and start to display. You can set a trigger for a voltage about halfway between the switch being open and closed, to catch the transition. Once you can see the switch transition, you can play around with the time base, volts per division, and trigger settings to better observe the noise.
Lastly as far as 'holding' the image on the display, your scope may or may not be able to do this. This type of functionality falls under the realm of storage scopes. Most new scopes purchased today are "Digital Storage Scopes" where the data is sampled, processed by a computer, and then displayed on a screen. Since the data is aquired and processed by a computer it is much easier to have a persistant display. This would be in comarison to most analog scopes, which may or may not have some form of storage functionality.