ていく and てくる: direction in space and time
Identify the reference point for movement or change. The same ending can describe a journey, a return, or a developing situation.
Move away from or toward a reference point
明日、学校に辞書を持っていきます。 says you will take a dictionary to school. ここに辞書を持ってきてください。 asks someone to bring one here.
The direction of いく or くる depends on the conversational reference point. Check the destination and speaking location even when the action 持つ stays the same.
An action followed by a return
パンを買ってきます。 often means you will go buy bread and come back. Include the return to the current location in your reading.
パンを買っていきます。 can mean buying bread and taking it along to another place. Translating both as simply buying bread loses the movement information.
Follow change through time
最近、寒くなってきました。 describes the weather becoming colder up to now. これから寒くなっていきます。 looks ahead to continuing change.
勉強を続けていきたいです。 expresses wanting to continue studying into the future. Time expressions help distinguish this use from physical movement.
Quick check: Does 少し飲み物を買ってきます normally suggest returning to the people you are with?
Show the answer and why
Yes. It normally means going to buy drinks and coming back. Here てくる combines the purchase with the return journey.