It is supposed to change.
Copper oxidises. Within a few weeks a new piece will start to warm and darken, and within a few months it will have a patina that is specific to you — where you hold it, how often, how humid your kitchen is. This is the material behaving correctly.
Everyday
Rinse with warm water and dry with a cloth. That is genuinely it. Do not put copper in a dishwasher — the detergent and the heat will strip it unevenly and leave it blotchy.
To bring the shine back
Half a lemon, a spoonful of salt, rub, rinse thoroughly, dry completely. Ketchup works too, for the same reason — it is acid and salt. Commercial copper polish is fine but unnecessary.
On the engraving
The recesses are meant to stay dark; that contrast is what makes the pattern read. When you polish, work with a cloth on the raised surface rather than scrubbing into the recess.
Drinking from it
Our vessels are unlined, which is traditional for copper drinkware. Use them for water. Do not store acidic drinks — citrus juice, vinegar-based drinks, wine — in copper for extended periods, and do not leave anything sitting in them overnight.