Hi there
I think you should get something like a focusrite Clarett or Scarlett with enough inputs. I'd probably go for 8. The Roland Studio Capture is another good option, but it has aged a bit (I have one).
Connect everything into the audio interface and record to a DAW like Reaper (many choices here). The only thing that might go elsewhere is midi from a keyboard.
Also figure out if you're gonna record or perform or both. Performance is more demanding in terms of space and the number of mics you need. Recording on a budget, you can overdub things. You might get by with less equipment and f.ex. wouldn't need the keyboard amp at all, though you should invest money into proper mics.
I might be able to help you out more, but I don't really post here much these days. Consider checking out the forum in my signature
Edit:
Also, the piezo pickups in ukuleles and guitars sound bad. It's preferable to record guitars etc. with a stereo mic setup (condenser mics since ribbon mics are more expensive). A popular budget option is a pair of Oktava MK-012 pencil mics, but again there are some options..