Hi rosebud123_,
Thanks for coming to Apple Support Communities for help today.
This article can help explain contacts on your iPhone: Manage and delete contacts on your iPhone, iPad, or iPod touch.
If you only want to start putting new contacts into iCloud, you only need to make it your default Contacts app account. Then you'd keep using your third-party account for old contacts. It sounds like you may have it set up this way now.
"Choose a default account to add new contacts
If you set up multiple accounts in the Contacts app, you can choose a default account to add new contacts to. Just go to Settings > Contacts > Default Account and choose an account."
If you want your third-party contacts to be synced in iCloud and then later remove the third-party account you will need to import them by following: Import or export a vCard in Contacts on iCloud.com.
Import a vCard
"If the vCard contains contact information for more than one person, each contact becomes a separate entry.
- In Contacts on iCloud.com, click
in the sidebar, then choose Import vCard. - Select a vCard to import.
Contacts from imported vCards are added to the All Contacts group. You can add contacts to any other group by dragging them. See Create and manage groups of contacts on iCloud.com."
Have a fantastic day!