It sounds like a server-client relationship (central database) and while SQLite is probably the simplest, note that its performance is probably the worst of the top 20 or so choices you have. It is different from Firebase and MySQL (and most other databases) in that it is embedded in the product, although it could be embedded in your server itself.
MySQL would require a separate MySQL db server, which means either two servers (one for MySQL, and one to provide your specific services to your client app) or both running on a single server machine. There are many alternatives in the same category as MySQL, and a choice of relational databases or document (NoSQL) databases. But architecturally, they are in the same category as MySQL, a separate db server that your application server would get its data from.
Firebase is different yet again, in that it is a service that is already hosted by a company, providing many integrated features such as authentication and storage of user account info. However it does take care of many of the concerns with running a server, such as performance, scalability and management. There are some negatives that you should be aware of though: any investment of time and coding with Firebase is pretty much non-portable, in that you are stuck with Firebase going forward. If you needed to switch to a different service, not only would it be a different API, but it would be a different architecture and much of your coding would need to be discarded. Second, it's owned and run by Google now, so you have a large corporation backing it, but that also means they could decide to discontinue it without any real effect on the Google bottom line. Also some folks would have concerns with storing data on Google servers. That said, I think if you are aware of these in advance, and especially if you are a high school student, that Firebase is a fairly easy winner here. The server is already set up for you, the documentation is very complete and rich, with lots of examples, and Google is not going away. The main concern would be if it really is massive, there could be a rising cost to the service. I suspect though that it is not massive, even if everyone in a school used it. The number of concurrent connections would not be huge (probably not even into the hundreds, even if there are thousands of users).
I'd go with Firebase even though you will need to learn their API, because you'll need to learn something one way or another. SQLite is a bit of a toy database, and MySQL is a real one but you (or someone) would need to manage that server on top of needing to develop the server and client app. With Firebase, much of the server already exists, including a professionally hosted database. There are tons of high-level features provided and initial cost is somewhere between very low and zero.
Part of this is dependent on what language you want to write this in. Javascript for a cross-platform client app (I'd use Vue.js + Vuetify for UI, and provide it as a web app and optionally wrap that with Electron for a desktop app, Apache Cordova for mobile). Server could be Javascript with an Express-based REST API on Node.js, talking to Firebase for services.
If you were a Java developer though, all this goes out the window and I'd recommend a simple Java server with Javalin for REST API, and embedded ObjectDB for database storage (combined into one server). ObjectDB is very very fast and can be separated out into a scalable server if this became truly massive. But you would probably never need to go that far.
All of this is a lot of work. I hope this isn't for something like an assignment. It is in the order of 6 months of work if you know what you're doing, all year if you're learning as you go.
