Why do ATC instructions to change level include the words "climb" or "descend"?
Adding the instruction "climb/descend" to the flight level is an example of linguistic redundancy offering a very rudimentary form of error detection for voice communication (which is inherently "unreliable", as it can be misheard by the receiver – on top of the sometimes poor ATC radio channel quality): 


  In […] telecommunication, error detection [is a] technique that
  enables reliable delivery of […] data over unreliable communication
  channels. Many communication channels are subject to channel noise,
  and thus errors may be introduced during transmission from the source
  to a receiver. Error detection techniques allow detecting such errors.
  (source)


Therefore adding the instruction "climb/descend" offers an additional layer of security, because


the pilot would never descend when instructed to climb (and vice versa)
and the pilot is able to detect an instruction mismatching the flight level, i.e. "climb to FL240" when already at FL320 (taking the example in your question)


... but not perfect security (because the pilot at FL320 would not detect an error if ATC mistakenly instructs him to "climb to FL420" instead of "climb do FL400"). Therefore ATC instructions are repeated on the receiving end by the pilot.

(For a non-aviation analogy, the same technique is applied when dates are agreed upon on the phone as "Friday, 4 January 2019", although the day of the week is already implied by the date itself.)

In essence, your question "Why do ATC instructions to change level include the words 'climb' or 'descend'?" can be best answered by the (limited) added information transmission security these instructions provide. Therefore for me the real question is why instructions concerning speed and heading don't obligatorily require this sort of disambiguating instruction – especially so as this unspecificity can demonstrably lead to confusion. I hope someone else will be able to answer this part of your question, as I don't have specific knowledge about this aspect of it.