The ADC can be set to auto-trigger from a variety of different sources. The best way to do it can depend on exactly what you're trying to do.
How often do you need take the sample? If it's very slow (like once per second) you can just poll it in code rather than fussing with interrupts.
Do these measurements need to be taken in close temporal proximity? Or can their readings be evenly spread out within the required time interval?
To do it exactly how you want it, I would broadly do it this way:
Start by setting up the ADC to trigger off of Timer1 overflow.
Set up Timer1 to overflow at the interval I want.
In the ADC ISR, keep a sequence counter variable that tracks which point of the reading sequence I'm on. On the first trigger of the sequence, change the ADC trigger to free-running, then make sure to set it at the proper point in the sequence so that the trigger source is Timer 1 overflow again after the last conversion. The sequence counter will also be necessary to determine which channel is being read from and which channel to set up for the next conversion.
Step 3 there will require a significant amount of thinking on your part to work out exactly how to juggle setting the trigger source so it is set back to Timer 1 at the right point in the sequence. You also have to take into account that in free-running mode the next conversion is already running, so even after you set the multiplexer in the ISR the next trigger will be for the old MUX settings. It's not super hard to work this out, but you will need to put your thinking cap on.
Do you mean "three analog readings from each of three different pins" (9 total readings) or "one analog reading from each of three different pins" (3 total readings)?
You can read one full-resolution ADC reading in just over 0.1 milliseconds so three total readings would be no problem but 9 would be a little tight.
"Quite often" is not precise. Exactly how often are you trying to measure each signal?
What are you measuring? If the signals are fairly independent of each other, there's probably no need to read all of them at once. Reading them one at a time at an interval might be good enough, and is probably easier to handle.
As for your other question, go to section 23.9.4 in the datasheet I linked to. The ADTS bits of the ADCSRB register allow you to set the auto-trigger source for the ADC. If you set it to autotrigger, the ADC will automatically start when one of those trigger sources happens. You won't have to do anything to manually start it, you just need to set of the ADC conversion interrupt to handle the result once it's finished.
