Вечером я пью вишнёвый компот дома.

I

Breakdown of Вечером я пью вишнёвый компот дома.

• пить: to drink
• вечер: the evening
• дома: at home

Questions & Answers about Вечером я пью вишнёвый компот дома.

Why is it вечером and not в вечер or в вечере? What case is вечером?

Вечером is the instrumental case of вечер (вечер → вечером). In Russian, the instrumental is often used adverbially to mean “in the evening / in the evening time” (same pattern as утром = in the morning, днём = in the daytime, ночью = at night).
You can also say в вечер only in very specific contexts (usually with an adjective/clarification, e.g., в тот вечер = that evening), but вечером is the standard neutral way.

Does Вечером mean “this evening” or “in the evenings (habitually)”?

By itself, вечером is often understood as a general time frame: in the evening (it can describe a routine or a typical time, depending on context).
If you want to clearly say “in the evenings (regularly)”, Russian often uses по вечерам:

• По вечерам я пью… = In the evenings (as a habit) I drink…
  If you want “this evening / tonight”, you’d typically say сегодня вечером.

Why is the verb пью (imperfective пить) used here? Could it be выпью?