What is the sampling-rate of the `analogRead()`?
The ADC in default configuration will take 104 µs to make a sample (which agrees with Edgar Bonet's comment of roughly 9600 per second).

You can speed that up somewhat by changing the ADC prescaler from the default of 128. See ADC conversion on the Arduino (analogRead). A prescaler of 32 will reduce that to 26 µs and still give quite acceptable results.

You can also run the ADC asynchronously (in the background) so you can transmit the previous result while obtaining the next one. A higher baud rate than 9600 baud would be highly recommended in this case.

People have been doing FFT analysis using the Arduino Uno and similar. I suggest you Google for relevant posts about that.