2013 Toyota Prius Three not III , strat up issue

Perhaps you can be more descriptive.
Having the engine run does change the sound of the car

The first minute of engine running will sound different because it's diverting its exhaust thru a heat exchanger to help the engine warm up quickly.

 

If it's knocking or running roughly and you did a very short cycle before (starting then shutting down the car less than a minute afterward) it's a known problem.
This happens particularly to people who have to start the car to move it in the driveway or garage briefly.

 

i think you're experiencing the cold start pre ignition as described by css28 above. just don't shut it off while it's still warming up, there is no other solution. many many threads here, going back to 2010, if you want to do some exploring.

 

don't start it up, vand shut it down until it is done warming up.

 

There is a service bulletin that the dealer can follow to replace the intake manifold with a newer design on the earlier Gen 3's but I think yours should already have the newer design.
If the car and weather are warm enough, see if you can put the car in EV mode for "driveway operations" when you turn it on. That will definitely avoid it.

 

How long does it take to warm up? I'm about to buy my first Prius.

 

it depends on the outside temp, and if you're calling for heat (which you can turn off. average is probably a couple minutes.