The way I saw it, «шутил» was potentially the past tense of «шутить». So, what's the reason for saying «пошутил»? Is it just to make the verb perfective? If so, does that apply to all verbs?
All I could find in my own research was that the prefix «по-» changes aspect in verbs of motion.
So, what is the reason for saying «пошутил»? Is it just changing the verb to perfective aspect?
Yes, adding the prefix по- changes the aspect of the verb.
It's not "just" changing the aspect, though. Different prefixes give different shades of meanings to the verb, or even completely different meanings altogether. Пошутил, нашутил, перешутил are all perfective verbs with different meanings taking different arguments.
There is no universal rule that would say exactly what prefix means what. Same as in English words of Latin origin like "compose, impose, expose, transpose, depose" etc, adding different prefixes to the same basic root spawns a plethora of different meanings, only vaguely, at best, related to each other and to the original root. You'll have to learn them separately.
If so, does that apply to all verbs?
In many cases, по- is the "default", so to speak, prefix, which is used when you need to put an unprefixed imperfective verb into the future tense without changing its basic meaning.
However, it doesn't work with all verbs (and, probably, not even with most verbs). The "default" perfective form can also be formed by suffixes (решить/решать), suppletion (говорить/сказать), other prefixes (писать/написать) or even, in the case of the so-called biaspectual verbs, by not changing the verb at all (женить, велеть, обещать).
In general, adding prefixes to imperfective verbs does make them perfective.
There is a class of iterative verbs formed with suffixes -ива, -ыва (шучивать, говаривать, глаживать etc.), which remain imperfective when prefixes are added to them. In fact, these suffixes are used to form imperfective forms back from perfective, and they don't keep the iterative meaning.
There is another class of verbs, called iterative verbs of motion (моторно-кратные): ходить, возить, носить etc.
When prefixed, they can form imperfective verbs too.
Sometimes, these prefixed forms have perfective homonyms or homoforms only differing in stress: заходить "start walking" perf., "visit, drop in" imperf.; выноси́ть "bring out" or "endure" imperf. / вы́носить "carry (a child in pregnancy)" perf, etc.
So, what is the reason for saying «пошутил»? Is it just changing the verb to perfective aspect?
Two meanings are added: the meaning expressed by the aspect and the meaning of the prefix. The prefixes can be different.
If so, does that apply to all verbs?
Almost. In motion verbs, по- doesn't always change the aspect. There are 15 pairs of motion verbs - of the same imperfective aspect, but different meanings (unidirectional verbs - multidirectional verbs): идти – ходить, бежать – бегать, брести – бродить, ехать – ездить, катить(ся) – катать(ся), лезть – лазить, лететь – летать, плыть – плавать, ползти – ползать, тащить(ся) – таскать(ся), гнать(ся) – гонять(ся), нести(сь) – носить(ся), везти – возить, вести – водить, садить – сажать.
The multidirectional verbs (the second example in the pair) sometimes, when prefixed, form a verb that can be used as a perfective and as an imperfective verb. And the perfective verb retains the basic meaning of motion, while the imperfective verb often acquires a different meaning.
