Get one list showing the way you dragged it before you add the category plugins. They solve a different job, and they are easier once this part is working.
Install Reorder Posts
In the admin, go to Plugins, Add New, and search for Reorder Posts. Install it and activate it.
Posts and Pages are already turned on. You can confirm that under Settings, Reorder Posts. The Reorder menu shows up for an account that can edit pages. An Author account will not see it.
Leave the other settings alone for this first pass. Attributes, and the sample query on the second tab, can wait.
Drag the list
Open Posts, then Reorder. For pages, open Pages, then Reorder.
Grab the dotted handle on a row and drop the row where you want it. There is no Save button. The new spot is stored as soon as you let go.
Pages can nest. The arrow opens the children, and you can drag a page under another page. That changes the parent as well as the order.
Screen Options at the top of that screen sets Items per Page. The default is 50. If a list has more than 1,000 published items, the screen warns you. Skip that list for now. A long list has its own guide.
Show that order on the site
A fresh install does not change the public site. The drag screen saved the order. The public list is still free to sort by date. Pick one of these two, for the list you care about.
If the list is a Query Loop block, select the block and turn off “Inherit query from template” if it is on. In the block settings, open the Reorder panel and turn on Custom Order. That uses the order you dragged, top to bottom.
If the list is the main blog or a post type archive, go to Settings, Reorder Posts. On the Settings tab, find Defaults for that post type. The type has to be enabled, or it will not be listed. Set Sort by to Menu Order, and Sort Order to ASC, so the top of the drag list is first on the site. Save Changes.
Sort by None leaves the theme or the block in charge. That is why the public page can look unchanged after a perfect drag.
If a theme or a custom query still sorts by date after this, use the next guide, Show the saved order on the front end.
Add categories, tags, or order inside a term
Both add-ons need Reorder Posts active. Install only the one that matches the job.
Categories, tags, and other terms use Reorder Terms. After you activate it, open Posts, then Reorder, and switch to the Reorder Terms tab. The same taxonomy can have a different order on each post type. This plugin does not change the public term list on its own.
The order of posts inside one category uses Reorder by Term. After you activate it, go to Tools, Build Post Term Data, and build once for the taxonomies you care about. Then the reorder screen has a Reorder by Term tab. That order is separate from the main list, and the public category archive will not follow it until a query asks for it.
Drag one list, look at the public page, and if the date order is still there, set Custom Order or Menu Order before you install anything else.