Fellow learner here.
Du and dig differ the same way I and me in English do.
Du is a subject, while dig is an object. For example:
Du ser musen. - You see the mouse.
Musen ser dig. - The mouse sees you.
Min and mit both mean 'my', but are used with different genders.
There are two genders of Danish nouns: common(-n one) and neuter(-t one). Articles, some pronouns and adjectives agree with the gender of the noun they determine.
So min kat, min mor, min bog.
And mit barn, mit vand, mit liv.
Same with din/dit (your).