In perfectly secret scheme, if key and message space are uniformly distributed, is ciphertext space always uniform as well?
Question: If we say that the message space and key space are uniformly
  distributed, this seems to me to imply that the ciphertext space must
  also be uniform. How would one go about proving this fact?


You don't because this statement is false.
Here's a simple counter-example:

Let $k,m\in\{0,1\}$ be a key and a message bit, both chosen uniformly at random. Then let the ciphertext $c=m\oplus k\parallel 0$, that is you do the standard OTP encryption and append a zero bit. Obviously this scheme is perfectly secret because the OTP is and the zero bit is completely independent of the key and the message. Also obviously the ciphertext doesn't have a uniform distribution from the set $\{00,01,10,11\}$ because $11$ cannot occur.

A more interesting (new!) question though would be though whether for each length-preserving perfectly-secret encryption scheme the ciphertext follows a uniform distribution.