Go here on a computer web browser: https://contacts.google.com/
Log into you Google account, see if your contacts have already been backed up.
If yes, then log in to the same google account on your new phone, contacts should sync.
If no, then your contacts are likely only stored on your old phone.
On your old phone, ignore the home button for now. Unplug it, then try holding down the power button for a long time until it reboots. Power button is usually on the side.
If your old phone reboots, then check the contacts app.
Assuming your contacts are still stored on the old phone memory:
- Connect the phone to WiFi
- Backup your contacts to your Google account (search online for guides on how to do this)
- On your new phone, sign in to your Google account and contacts will sync across
It's better to backup contacts this way, as if you ever lose your phone or SIM, your contacts are always backed up safely.