This is a frequent problem for quite a few of the customers I support.
Firstly, make sure you know your email address and password then log on to the web servers. You need to understand, when you send and receive emails, they are actually delivered to large computers (servers) somewhere on the internet. These are managed and run by your email provider. Your computer has an app that talks to these servers, so what you see on your email programme on your PC is actually a display of what is on the servers. You have two ways of accessing emails, one via the app on your PC and the second by logging onto the website (and subsequently the email servers themselves) of your email provider. I'm explaining this so you understand that, if something goes wrong with your PC, you can still access your emails from another device by logging on to the website of your email provider (e.g. if you use gmail, google ‘gmail login’ from your browser search bar). You will need to enter your email address followed by your password when requested. If you have forgotten your password, at this point you can click the 'Forgot password?' link and you will get an email on your PC with a link and instructions to reset your password. (When you reset your password, don't forget to change it on your PC email programme or else you will only be able to access your emails from the website.)
Once you've logged into your email servers, you will be able to check your saved contacts too and your account settings and set up folders for filing mails if you want to. (Beware, security has tightened up considerably and you may need to give a mobile phone number which will receive a code that needs to be entered whenever you log into the website servers.) If you have saved important emails on your PC and are worried about losing them, you could print them off or forward them to yourself so they appear on the servers again and you can file them again on the new PC.
As for forgotten passwords, I have to say that is is better not to ask your PC to remember passwords as you will forget them (as you have). Using a password manager is good advice. You can reset your passwords by going through a similar process as for email. If you must write passwords down somewhere, remember to be precise, upper case, lower case, spaces etc. all count. The same goes for security questions. Don't write 'Park Street' if you entered 'Park St'.
As for changing operating systems, I'm afraid it is something you won't be able to avoid and they are all a bit different. You will have a learning curve, whichever PC you choose. My advice is to make YouTube your best friend. If you want to know how to do something, just query on YouTube but make sure you say which operating system you are using.
Finally, if you have a good broadband link and you don't really have a particular, high specification, computing need such as gaming or video editing, you might also consider a Chromebook. You won't ever have to upgrade the operating system or worry about antivirus. Like cofusedbeetle, I use Windows 10, Apple and Android but I have found that customers with a Chromebook seem to have fewer problems with their computers once they have the hang of using the programmes on it. Get plenty of advice and try various models and operating systems.