Jumping from microcontroller to DSPs
[Full disclosure: I am a TI employee who works in the C2000 group.]

The Piccolo series, like all C2000 products, are microcontrollers that use a C28x DSP for their CPU. Although it has DSP features and architecture, the C28x also does everything that a general-purpose CPU does. Using a C2000 MCU is basically the same as using MSP430, TMS570, or any other MCU -- you write code in C, program it to the flash using TI-provided or third-party tools, and let it run.

For something like the C55x series, which has no on-board flash, your code needs to go in an external memory. The on-chip boot ROM has options for loading code from serial peripherals or from an external memory interface. That's about all I can tell you there; I don't know much about C55x.