How to measure output of the ultrasonic transducer?
There are three main aspects to working with ultrasonic measurements:


Generating a tone at the resonant frequency of your transducer
Receiving the tone through a band-pass filter at the resonant frequency of your transducer
Measuring the time between the sending of the tone and the reception of the echo.


The circuit you have found there is overly simplistic and doesn't do everything you need.  You have to generate the tone at the right frequency with the Arduino (just about possible with manual tweaking of the PWM module), and it can receive and amplify audio through the transducer.  What it doesn't do is filter than input to isolate the tone from any background noise.  It just receives and amplifies anything.  Yes, the resonance of the transducer does provide a certain amount of filtering, but isn't very good by itself.

Also the amplifier show is just a placeholder. There's a lot more circuitry needed with it to make it work.

And then you have to measure the time.

All in all it's a lot of work.  Too much work for a beginner.  You would be much better off using an ultrasonic sensor module that has all the hard work done for you.  I can't recommend a module since you don't say what kind of range you are looking to achieve.