Android: The built-in dialler for your phone varies based on who made it, but Simpler Contacts is available for everyone and offers plenty of features, including the ability to back up or export your contacts.
Most Android users probably use Google to backup their contacts, which is fine for swapping phones. as long as you're not changing platforms. Simpler, on the other hand, allows you to export your entire contact list as a .vcf file, which can then be imported to Outlook, iCloud or any contact manager that supports vCard.
As a standalone dialler, Simpler is still pretty great. It's designed with Google's latest visual style in mind (which is nice since most phones don't have 5.0, and the dialler is one of the few apps left that requires a system update to get). It also has useful features like search — both text and T9 style — and a wizard to merge contacts.