Q: If I set up my phone as "new" will it delete my phone numbers?

Greetings ultimasfirst,

There are a number of ways to sync your iPhone's contacts, including using iCloud, which is what it sounds like the person at the store did. See this article -

iCloud: Change iCloud feature settings

There are other methods. See these articles -

Sync your iPhone, iPad, and iPod with iTunes using USB - Apple Support

Sync your iPhone, iPad and iPod touch with iTunes using Wi-Fi - Apple Support

Syncing contacts specifically is described in this article -

iTunes 12 for Mac: Sync contacts, calendars, and other info with iPod, iPhone, or iPad

After setting up an iPhone as a new device, it is in its factory configuration, as if it just came off the manufacturing line. Your phone numbers and all other data would NOT be on the phone. See this article for reference -

How to erase your iOS device and then set it up as a new device or restore it from backups - Apple Support

Thanks for using Apple Support Communities.

Be well,

Brett L