Achieve – Succeed by working hard: achieve success, achieve goals, objectives, progress, we achieve results

*General & common 

She achieved first place in the race.
I achieved a measure of success in high school.
He achieved his dream of becoming a doctor.

Accomplish – Complete a project successfully: get (a task) done; accomplish goals, tasks, and missions

*Sounds good with things related to productivity.

They accomplished their project on time.
She accomplished everything on her to-do list.
He accomplished his goal of writing a book.

Attain – Reach a level, rank, or designation: this is formal language – not everyday speech.

*Sounds good with abstract ideas and high standards.

He attained a high level in English.
She attained the rank of manager,
They attained financial independence.
The product attained international safety standards.

Summary: Achieve is the most common and general. We can achieve and accomplish goals, we accomplish tasks, and we attain high standards or abstract levels, formally.