Welcome to the wonderful world of SharePoint Development!
You can absolutely create a form using the SharePoint Framework that lives on it's own page (web part) and saves/loads data from SharePoint lists.
TypeScript is all you need (just select no framework in the generator). However, many find it easier to take advantage of things like React where there are several libraries and samples already available.
I'd also suggest looking into PnPJS. You can communicate back to SharePoint using the SPHttpClient within SPFx no problem, but I find PnPJS makes it much easier.
Finally, there is a lot of information out there and it can be overwhelming. I highly recommend starting with some of the available samples from SharePoint Patterns and Practices (PnP). These are full projects that illustrate how to do things and you can often adapt a sample to meet your needs or at least see how real world devs already tackled a similar problem.
Here are some samples that you may find helpful:
From a less developer focused perspective, you might look into Microsoft Forms (depending on how custom you really need to get). There are available templates from Microsoft Flow to automatically save results from a form submission directly into a SharePoint List. This is definitely a one way (so only submission) approach but the form itself can be hosted on a modern page or embedded nearly anywhere else.
Moving up in difficulty but still less developery is PowerApps. Generating a quick PowerApp from an existing list is pretty painless and then it can be customized from there. You can also host your PowerApp on a modern page.