How to control a brushless DC motor without an ESC with Arduino

You should definitely use the ESC. Brushless motors works best when driven with a sine wave (or as close as possible to a sine wave). They also require a fairly accurate and complicated set of signals. Generating the proper wave forms and timing from an arduino would be difficult, and unless you really need to it, is probably not worth it. You can always arrange to control the ESC from your arduino, which would give you programmatic control plus the efficiency and power of the ESC.

If it's the kind of ESC used with R/C planes, the hook-up is pretty straight-forward.

On the output side, there will be three wires to connect to the motor. It almost doesn't matter how you connect them, except that if your rotation is opposite to what you expected, you can just swap any two of the three.

For input, the ESC takes a power feed from your DC source, usually a lithium-poly with planes, and a control signal from the R/C receiver.

The control signal is the same kind used with servos. There is a pulse of a certain width, and variations in that width cause variations in servo position, or in the case of the ESC, the resulting speed. Point being, that if you can get control an ordinary servo with your Arduino, then you have practically everything in hand for controlling an ESC.