Transfering the site

Hi everybody. I have a wordpress site on a site of my job. We are about to transfer our domain and hosting to another company this week so i want to backup somehow the actual running wordpress site, settings, pages, databases, etc so that when i complete the domain transfer i can do a restore and have the site running again with the same looking and functionality as before.
Someone please help me, i tried “DUPLICATOR” but when i see the site it’s messed up, the pictures are not aligned or anything, it just displays everything really bad.
Does anybody know how can i save the actual wp site and it’s database so that when i transfer the domain to my own hosting company i can upload all this wp settings easily?