Yes, this is a very common phenomenon - it's called Palatalization. It happens when a consonant is followed by a front vowel (e,i, etc). However:
it does not happen to all consonants - depending on the language and its evolution only some consonants may be affected. (as in your example, k does not become ch)
it does not happen with all front vowels. In Japanese it's only before
This can be found, in various variations, in many languages.
This explains only the change before "i". The explanation for tsu can be found in this article
/t, d, n/ are laminal denti-alveolar (that is, the blade of the tongue contacts the back of the upper teeth and the front part of the alveolar ridge) and /s z/ are laminal alveolar. Before /i/, these sounds are alveolo-palatal ([tɕ (d)ʑ n̠ʲ ɕ (d)ʑ]) and before /u͍/ they are alveolar ([ts (d)z n s (d)z]).
The latter phenomenon is rarer (I can't think of examples other than Japanese). But consonant changes that depend on the preceding or the following vowel are common.