WordPress Move

I am new to wordpress customization and Im not technically inclined, so bear with me. I customized a WordPress website for a school with an existing website and domain. The site was customized at webhost A, while the old site, (made with microsoft page) and domain reside at webhost B. I need to migrate the new site to webhost B and to install the new site at the existing domain hosted at webhost B. I installed WordPress Move in the back end of the new site to make the migration easier (I hope). Please outline the steps I need to take (because based on my reading you can make missteps with WP Move) to prepare for migration. I dont want to lose access to the site, and I dont want the URLs messed up. Also, is there a delay in the site showing online once the migration has been completed?