Minimal Arduino using ATmega32U4
Several Arduinos and Arduino-compatible boards use the ATmega32U4.
Many of them have published open-source plans that show exactly what components are used and how they are connected. They may not be the absolute minimal number of components, but it seems clear that you don't need more components than a known-working board:


1 ATmega32u4
1 16 MHz crystal
2 22pF ceramic caps (for crystal)
1 uF decoupling cap
1 pushbutton
1 10 KOhm pull-up resistor (for pushbutton)
1 LED
1 220 Ohm resistor (for LED)
1 USB socket
2 22 Ohm resistors (between USB socket and ATmega32u4)


Alas, there are two unnecessarily difficult hurdles between newbies and getting a new ATmega32u4 device connected over USB to a computer:


The ATmega32u4 and nearly every other device that communicates over USB is only available in SMT packages with pins at 0.8 mm pitch or even closer together.
Most people find packages with pins at 2.54 mm pitch easier to handle and plug into a breadboard.
Installing a bootloader into any new processor chip is often tricky.


As Russel already pointed out, you can save a lot of time by starting out with an Arduino or Arduino-compatible board that already has the chip soldered on and the bootloader programmed in, bypassing the above unnecessarily difficult hurdles.

A few Arduinos and Arduino-compatibles (with links to schematics, etc.) that use the ATmega32u4:


Bootloading an ATmega32u4 with Arduino
Teensy 2.0 (ATmega32u4 USB dev board)
Arduino Leonardo; Arduino Leonardo
Arduino MICRO
SparkFun Pro Micro (Arduino-compatible)
FLORA - Wearable electronic platform: Arduino-compatible
Adafruit Feather 32u4 Basic Proto
Installing Arduino Bootloader on an ATmega32u4; Installing Arduino Bootloader on an ATmega32u4