Simple "Computer Design" project
I'm not sure how far you want to go with this, but I'm afraid you may become frustrated when you find out how complex your design may become if you want it do something useful, no matter how nice and comprehensive it looked when you started.
Therefore I'm back with the suggestion to your previous question: the FPGA. There are complete so-called "soft processor" designs available free, but you can also build your own, starting from a 2-bit full adder if you want. (And that is what you want if you still have to learn VHDL.) Every FPGA manufacturer has development boards for its products, and there are also third-party boards, with a choice of on-board peripherals, like LEDs, character LCD, buttons, etc.