Recommendations for DSP Hardware
Since what interests you is the "embedded system" part, and since you have a low budget (this excludes anything that requires proprietary compilers), I'd recommend building yourself a board with an ARM MCU and a codec, like this one. There's less than $50 of parts - the processor, the codec and the bare minimum to get them to work.

I'm recommending this because you can easily grab a FOSS toolchain for arm-none-eabi, because these chips have a serial bootloader which allows them to be programmed with a $2 FTDI dongle, and because they have enough flash and RAM onboard for many applications - so you don't have to deal with the complications of an external RAM or flash chip. For a beginner, it's good to have a small self-contained, system you can entirely master!

You'll learn how to set up a toolchain for an embedded processor, how to use a JTAG or serial interface to flash the chip, how to configure the peripherals and hardware for your application, and you'll write code in a relatively basic environment. It'll be a fun experience! The STM32F4 is good enough to implement audio effects and 6-10 voices of audio synthesis (Used in: the Owl stompbox, Audio Damage and Make Noise Eurorack modules, DIY synth projects like PreenFM2, Audiothingies P6, or Sonic Potions LXR). UAD needs a lot of computing power because they emulate analog hardware at the component level. You are not going to do that. To give you an idea of what's possible, the STM32F4 can run about 250 biquad filters or 100 wavetable oscillators or a phase vocoder with 75% overlap. This is a general purpose architecture, but there are DSP-ish things like saturation or MAC instructions that make the generated code relatively compact on audio DSP applications.

What this won't teach you are the specifics of VLIW architectures, and the art of writing assembly for such architectures. For that, you'd need a proper DSP board from AD or TI, but this costs more, and once you build something with their devboards it's more expensive and complicated to replicate it on your own board. You can partly capture this experience with a dsPIC.

Another cheap toy you could play with: ADAU1701 board. It has a built-in AD/DA. Downside: it can only be programmed through AD's drag and drop tool, with building blocks like filters, dynamic processors... Elektor also has a spin FV-1 board - it's a very peculiar minimalistic chip that can be mastered in a few hours. Might be worth exploring if you like vintage reverbs.