For example when you go to get new job, you will be given a "тестовое задание". How to translate this combination of words into English?
This type of question is definitely off-topic on this site. However to help you out, the right translation is test assignment or trial assignment. I work as a manager in the software development organisation in England. When I look for new people, part of the selection process is technical test.
Once a candidate is hired, they will typically be given some test assignments (in my previous job we called them trial assignments) to confirm that they can really do the work expected of them.