Drupal has some mainstream contributed modules that can handle this in most cases. Using built-in interface translation and i18n_string, you can add translations for many of the main breadcrumbs in Configuration -> Regional & Language -> Translate Interface -> Translate.
Several modules can help you get these strings to begin with. You can use the potx module to pull all built-in interface strings from your other installed modules, which puts the strings into the locales_source table. Then you can use the l10n_update module to search for community-contributed translations for these strings.
For the last part of a breadcrumb that comes from a node, for example, Drupal will display the translated title of the node if it finds it. For Content Translation (node-based), the node displays its title in the breadcrumb, and for Entity Translation (field-based), you can enable the title module to convert the title property to a field, which can then be translated and displayed in the breadcrumb.