The border between DSP and microcontroller has blurred
considerably in recent years. Many microcontrollers now have DSP
functions (single cycle multiply accumulate), and some DSP's now
have RISC instructions sets and microcontroller-style
architectures. Also, it's common to program DSP and
microcontroller functions into gate arrays such as Xilinx.
Frankly, if I was starting today to learn about DSP's, I'd
seriously look at becoming familiar with FPGA's, since that
gives you a lot of flexibility with processing and interfaces.
Maybe look first at the Xilinx Spartan 3e FPGA as a useful point
of reference.