I have the most experience with using PIC micrcontroller.
There are essentially two different USB "super classes", USB device and USB host.
Consider the analogy of a USB mouse plugged into a PC. The USB mouse is a USB device and the PC is a USB host.
A USB device can be implemented as several different classes: Communications Device (CDC), Human Interface Device (HID), Mass Storage Device (MSC or UMS), are some of the more common.
The Microchip USB development board previously posted, can implement the above device classes.
However, if you wish to implement a Host class device, you'll need to consider either PIC24 or PIC32, with the appropriate USB peripheral module, based board .
Related to this question would be, what is a good board to get introduction to ARM class devices?
There are numerous ARM development boards available with an ARM which offers a USB peripheral module.
Some of the more popular are the NXP LPC17xx, TI Stellaris and Atmel SAMx series ARMs.
You can find an NXP LPC1768 or LPC1769 based development board in the $20 to $80 USD range, depending on the desired extras.
Also search eBay for an abundance of Chinese clones of popular dev boards like the KEIL MCB1700.
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Based on your familiarity of PIC, I would recommend starting with a dev board based on that family and then migrating to an ARM based dev board.
BigDog