It seems like every other question posted nowadays is about how to upgrade an older version of oscommerce and the op usually references the official version. This post is meant to provide a quick reference for how to convert an older shop to the current, recommended, one.
First, the version that should be used is commonly called the Bootstrap, BS or Responsive version. It can be located here. One should not use the official version. The official version can be installed via the hosts control panel so it is easy to think that is the one to use - it is not.
Since the BS version is not directly compatible with other versions, the following steps will be needed:
Install the BS version into a new location
Convert the database. See this thread for help with that.
Install whatever addons the site needs. You need to be sure the addon will work with the BS version. This thread mentions specific ones but there are many others that will work. If in doubt, ask in the support thread for the addon in question.
If you intend to use paypal, then install the Paypal App.
At this point you should have a working shop. Test it thoroughly, especially the payment and shipping modules, before going live with it.
When you are ready to go live,
Mark your current shop as down for maintenance ( a simple index.html file is fine for this purpose ).
Make a backup of the current shops database and files.
Convert the database to the BS version.
Replace all of the files with the new shops files.
Edit the configure files so they work as the live shop.
Remove the blocking page.
If everything has gone correctly, the conversion should be complete.