I have completed my profile including picture, bio, skills and past experience. Should I wait for a verification e-mail or something like that to start working ?
You will get a confirmation email from Upwork within 24 hours. In the mail, they will either accept your application or decline it. If they accept your application, you can start bidding from then on. But by any chance, they decline it, you have to check out your profile, update it again and you need to submit your profile again for approval.