You can run the Arduino IDE on Windows, Linux or Mac. I use Linux and I suspect it is marginally easier than Windows (assuming you know Linux). The Arduino IDE is written in Java.
The Arduino Servo library produces the proper pulses needed to control servos and ESCs. For example myservo.write(90) should move the servo arm to 90 degrees. You can also use myservo.writeMicroseconds(1500) to do the same thing.
The Arduino can send control signals to the ESC but does not replace the ESC. An Arduino cannot handle the high currents and voltages needed by motors. An alternative to using an ESC might be one of the motor shields. But if you already have an ESC I would use that.
You can study Arduino programming using the IDE before you buy an Arduino board. The IDE has many useful example programs (sketches in Arduino speak). There is also a huge amount of online assistance.
The Uno is probably the best board for a beginner as it is the most standard of the Arduino boards.
...R