You haven't specified one of the more important ADC parameters, which is the number of bits of resolution. This selects the magnitude of your quantization error. ADC's typically come in 8, 10, 12 and 16 bits. 16-bit ADC's are fairly rare. So I looked for chips with 12-bit ADCs.
A sampling rate of 8000 samples/sec should not be an issue. For example, the Microchip PIC24EP256GP202, which costs just $5.41 in single quantities from Digi-Key, has a multi-channel 12-bit ADC that can sample at 500,000 samples/second, obviously way more than you need.
It also has 256KB of flash and 32KB of RAM, and two UARTs, so you can use one to connect to the WT-12 (you just need to use cross connected the TX and RX pins between the WT-12 and the PIC), and use the other if desired to connect to a terminal program on the PC for debugging output.
Because of your proposed sampling rate of 8000 samples/sec, if you use two bytes for each 12-bit sample, this would require a serial bandwidth of 8000 * 2 * 10 bits (for 8N1 serial protocol), or 160,000 bps. This is obviously higher than the default baud rate of 115,200 for the WT-12. However the WT-12 can run up to 3M baud, and the PIC24 can go higher than that. So this is not a problem after all. So I suggest picking a baud rate of 460,800.
If you chose to use an 8-bit ADC (which I do not recommend), you could get by with the default 115,200 since you would only be sending one byte of data, or half the bandwidth of the above (80,000).
This PIC24 is available in a DIP package, which makes it easier to prototype with.
For programming, you will need to get a PIC-compatible programmer, such as the PIC kit3. (This is the least expensive one made by Microchip; there may be some third-party ones you can also buy.)
There is a free compiler available for the PIC24 that is free from Microchip; the only difference between the free one and their paid one ($495) is the free one doesn't do any code optimization. But since the processor runs at 70 MIPS, that won't be an issue in your case. You will also need to download the MPLABX IDE, which is free.