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 i
.
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.