Update packages conflicting with core update
Update:
Thanks to the comment below from @mpdonadio below, it seems better to follow the official documentation right here: https://www.drupal.org/node/2700999#s-special-considerations-for-upgrading-to-drupal-880-and-later

composer update
git add composer.lock; git commit # Save the updates as a separate commit.
composer remove webflo/drupal-core-strict --no-update
composer remove drupal/core --no-update
composer require 'composer/installers:^1.7' --no-update
rm composer.lock
rm -rf vendor # Also helps avoid conflicts.
composer require drupal/core-recommended:^8.8 --update-with-dependencies
git add composer.json composer.lock; git commit -m "Update Drupal to 8.8.0 and use drupal/core-recommended instead of webflo/drupal-core-strict"




Old answer

First, make sure your composer version is not fixed (e.g. drupal/core:8.7.10):

composer require drupal/core:^8.7.10


This will update to its latest version (e.g. 8.7.11) but also makes sure you can progress later. To see which projects would cause problems, run:

composer prohibits drupal/core:8.8.0


This will result in a list:

typo3/phar-stream-wrapper (^3.1.3)
typo3/phar-stream-wrapper (but v2.1.4 is installed)


Then you can make sure you update the package:

composer require typo3/phar-stream-wrapper:^3.1.3


Then update like this:

 composer update drupal/core typo3/phar-stream-wrapper pear/archive_tar