What is the best way to solder these wires to circuit board?

If you do want to solder them directly, you can make it more robust (and marginally better looking) by drilling holes wide enough for the wire + insulation to fit snugly, looping the wires through those holes and then soldering them. This provides strain relief so that the solder joint isn't taking a significant mechanical load.

This is illustrated below, but you can also have the holes closer together, the other way round, etc - whatever's convenient. Image credit: Windell Oskay, CC-BY

You could solder 0.1" pitch headers onto the board. Then use crimp terminals on a mating connector.

Molex KK connectors are a common type, but there are many compatibles from other manufacturers. You'll need a suitable crimping tool, but after that investment the costs are low.

You could try Configurable Spring Terminal Blocks from Adafruit Industries http://www.adafruit.com/product/1074 or 2/3 pin- 3.5mm Terminal Blocks http://www.adafruit.com/product/724 that snap together. I have used these on Arduino projects and the work nicely.