Help with logistics of my new site

Hello everyone. My question is going to be a bit involved, so I want to say now that I highly appreciate each and every response I get. Here it goes:

I downloaded WordPress for the first time just a few hours ago and uploaded a theme that I purchased from a third-party. Now I’m conflicted as to what my next move should be. My site architecture should work as follows. https://www.mysite.com should direct to a nice and simple page with a search bar (I don’t think this page should require WordPress to create. I think I can just code it myself). In this search bar, the user will type in his/her university. As they type, the options will narrow down to their choice. Once they click on their choice, they will be directed to a subdomain. For example, if his/her school is Boston University, they will be directed to bu.mysite.com. Each school site (and each subdomain) will contain unique data, but will use exactly the same features included in the theme that I purchased. Luckily, my theme license allows me to install the theme on an unlimited number of sites. How should I go about accomplishing this task? Where should I install WordPress? I appreciate any tips I can get. Thank you!

Btw, I’m new to WordPress and making websites in general.