Downloading Drupal and installing the files (GUI)
I. Download and unzip the latest Drupal release to your desktop.
You can download drupal at the Downloads page. You can use unzipping software such as WinZip or Stuffit.
II. Move your old Drupal files
You made a backup, right? Using your FTP client, move all your old Drupal files on your server to backup directory. If you are upgrading a test site, move your test site files. It is considered best practice to remove the old files completely rather than overwriting to ensure you have a clean installation of the new files.
III. Upload the new version of Drupal
Using your FTP client, upload the new version of Drupal to the same place where your old files were located. If you are upgrading a test site, upload the files to your test site directory.
Make sure your files directory is writable.
IV. Upload contributed modules
Again with your FTP client, upload the new version of all your contributed modules. Make sure you have the version of the module which matches your new Drupal version.
V. Copy over necessary files from the 'backup' copy directory
Using your FTP client, copy the following files from your 'backup' directory to the Drupal directory on your server:
- .htaccess
- sites/default/settings.php (but see section on older sites below)
- the 'files' directory
- any other files you need from the 'backup' directory, such as your cron scripts in scripts/ or subdomain folders, or any customized themes within sites/all/themes/custom/
Older sites
If you are upgrading from a version of Drupal that is older than Drupal 4.5, you will have to look in your 'backup' directory for a file called:
<strong>includes/conf.php</strong>
Then using a plain text editor such as jEdit, copy the following three lines as they appear in your conf.php file:
<strong>$db_url = mysql://user:pass@localhost/drupal_db';
$base_url = 'http://www.example.com';
$db_prefix = '';</strong>And paste them over the appropriate lines in your new Drupal file on your server, located here:
<strong>sites/default/settings.php</strong>
It's simply a matter of overwriting the above three lines in settings.php with the lines in conf.php.
These settings are responsible for connecting your Drupal to the database and to the files it needs. The file name and location was changed from Drupal 4.5 to 4.6, hence the magical dance involved.

Are these instructions correct for 5.2 -> 5.3?
The upgrade instructions (here, and similarly on the command line page) include:
Elsewhere, I saw an update to .htaccess listed as one of the changes from 5.2 to 5.3.
I presume I should not copy the old .htaccess over the 5.3 one?
Yes -- .htaccess changed
Yes -- .htaccess changed from 5.2 to 5.3.
If you've made changes of your own to it, you need to examine your copy and make the same modifications to the new one.