The chances are your Arduino wouldn't have enough memory to efficiently write a driver to control the USB for one webcam let alone be able to perform all the functions you require. The Raspberry Pi is effectively a small all in one PC for around £25 which will run Linux, has plenty of accessible I/O, a network port, a HDMI socket, it can support full 1080p video, two USB ports, onboard camera port, SD card socket and much more.
Do a Google on it and have a read through - great for anything that doesn't requite too much processing power.