In the Android world, contacts on your phone automatically back up to a Google account, so coming from another Android phone, you should be able to access your contacts by associating your Google account to your Android phone -- one of the first steps when setting up a new phone.
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If, however, you need to import contacts from your SIM card, it's a simple process. Once you've finished the import process, go to "Accounts" in Settings to make sure all your contacts, including new ones, sync to your Google account. Before you start importing, create a Google account if you do not yet have one.
Importing Contacts From Your SIM Card
Open your "Contacts" app. Tap the menu icon next to the search field and select "Settings." Some devices running Android 5.0 or later have "Import/Export" on this menu. If that's the case, then select "Import/Export and proceed to Step 4.
Choose "Import/Export Contacts" from the Contacts Settings.
Tap "Import From SIM Card."
Choose the Google account for backing up your contacts. Your contacts are then available, not only on your phone, but in the "Contacts" section of Gmail in your browser.
Select "Import All vCard Files," which is simply another way to tell your phone to "Import All Contacts." The information for each contact is stored in a VCF file or "vCard." After you select this option, a status bar pops up showing the progress of the import until it completes.
Associating Your Contacts to Sync With Your Google Account
Select "Accounts" from your phone's Settings app.
Select "Google." You can associate your phone with more than one kind of account. Many accounts are associated only with specific apps.
Select your Google account.
Check the box next to "Gmail."