Issues using HC-06 Bluetooth module with arduino
I assume this wiring is correct, right?

Rx &gt; Tx
Tx &gt; Rx
Gnd &gt; Gnd
Vcc &gt; 3.3V



Almost. The Arduino's TX should be reduced to 3.3V using a 10k/20k potential divider.


  How do I read the serial input? I've been using

if(Serial.available() &gt; 0)
       phoneInput = (char) Serial.read();



You should read and digest this.


  What datatype is the Serial.read() returning to my variable before I cast char on to it?


An integer. It holds either 0-255 representing the ASCII code for a character, or -1 if there is nothing in the buffer to read.


  I get errors uploading the code to the board (avrdude: stk500_recv(): programmer is not responding), until I remove the pin connected to the Rx on the arduino, is there a cleaner solution to uploading the board without unplugging wires?


That will be because you are using pins 0/1 to communicate with the bluetooth which you can't do because those are used to communicate with the PC.