I just started learning Flutter and Dart a month ago with the hopes of writing a web app. Here's something I wish had been made more clear when I first looked into it:
Flutter is not yet ready for releasable web apps.
Web functionality is currently on the "Developer"-level channel. That's one level above the cutting-edge "Master" channel, and one level below "Beta", which is one level below "Stable".
I know, you watch the official videos and read the pitches and they say "Mobile AND WEB" but it's just not true yet. It's obviously the goal, but the dev team has put more emphasis on making it mobile (iOS and Android) ready. No digs against them; it's just where the focus is being put.
To give you an example, look at this FlutterFire page and notice that there's a separate column just titled "Web?" for all those plugins. Until very recently, there were only two plugins that were officially web-ready. And the other two of the four there are so new at the time of this writing they're not real-world tested enough yet to be stable.
A more to-the-point example of Flutter's shortcomings on the web as it pertains to just the UI portion of it: when you build a screen with more content than will fit on one screen, you will not get a scroll bar in your web browser. Your user will have to click the mouse on your content and drag it upwards (like their finger on a mobile screen) in order to scroll down.
If you really are a COMPLETE beginner, then depending on how much time you take, many of the functionalities that you're looking for may very well be released just before you get to them. But that's the optimism that I initially had for myself as a non-beginner, and it ran me into a lot of dead-ends and wild goose chases when I'd learn about a functionality that works in mobile, but does not work in web.
All that being said, IMHO Flutter has enormous potential to live up to the cross-platform hype. And I'm not abandoning my project; I'm just accepting that I'll have to be a bit more patient with the pace of Flutter's evolution and with myself trying to learn it.
And depending on what exactly you want to do with your app, the developer-level web app functionalities that are already pretty stable may be all that you need.
So if you can get away with developing your app on just one mobile platform first, and spend your energy learning the environment as a whole instead of figuring out how to make Web work, then by the time you got to the polishing stages of your app, if it's simple enough then Flutter will be able to publish it as a Web app for you as well with minimal tweaking.
Good luck, my friend!