How to measure AC and DC with oscilloscope
First of all:

Yes. All the ground probes of your different channels are connected internally. This means you can't connect grounds a different voltage levels because they're shorting on the inside of the oscilloscope.

Second:

Your ground probe from each channel is loosely connected to the earth wire from the oscilloscope's plug. This means that you should never use the oscilloscope to measure the voltage from the wall unless you're using an isolation transformer. Otherwise you risk frying it and/or activating the differential protection.

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For what you have it seems that it would be more interesting to see the waveforms at VI and VO. This share a common ground so you can see them both at the  same time. Don't use AC coupling for this, since it removes all the DC component which is what we're trying to see at the output. 

Start with that and then see what happens when you increase/decrease the input/output capacitance. Learning by doing will work wonders.