Upgrading Drupal

Hello - I am a user, not developer, and since the person who set up my site five years ago stopped working with Drupal earlier this year I am having nothing but problems. I am trying to migrate from 7 to 8 (and then hopefully 9). I have employed a specialist company to do this but, after four months, still don't have the site working as seamlessly as previously and it is having a financial impact on my business. As I am not a developer I am struggling to understand why this is the case.

Starting from the beginning, I have been a happy user of Drupal for over 10 years, starting with Drupal6. I have never used Composer or Drush. All installs have been done either via ftp or CPanel. Many times I have read up on Composer but just haven't got my head around how it works, but increasingly I am finding modules now have to be installed via Composer meaning that I can't use them. Converting to Composer is way beyond me and I suspect others as well.

We've upgraded four old sites from Drupal 8 to 9 so far and I thought I'd share my notes with the community. I can't guarantee this will allow you to upgrade seamlessly, but my hope is that this thread will come up when folks do keyword searches for specific problems and it'll at least lead them down the right path.

I'm taking over the development of a site that is stuck at version D9.1.10.
"Available updates" suggests me to update to version D9.2.6, but in the new version page there is the note "Sites on 9.1.x or earlier should update immediately to Drupal 9.1.13 instead of this release".

I'm a little unsure what to do.
Why this note...are there incompatibilities between the two versions?
Which version can I install, considering that my host uses Php version 7.4.13?

Thanks to whoever can give me some indications

I'm making progress on getting composer based install and update to work on shared hosting.  Install works, update of the drupal core works.  Beautiful thing.

My directory structure is

public_html

 ... web

 ... vendor

the composer.json file is in the public_html directory and I run composer right from that directory.  Earlier today 9/16/2021 I ran:

composer update drupal/core "drupal/core-*" --with-all-dependencies

Hello,

I try to upgrade my Drupal 8 website (version 8.9.18) to Drupal 9 (version 9.2.5) and i get the following errors :

First try