Network issue

1.

Remove and reinsert the SIM card.

If it doesn't help visit your carrier, maybe the SIM card is damaged, ask for a replaced (new) card. Normally it is free.

Ask also to check if the SIM tray is bent, as even a tiny misalignment can cause the SIM card to not fully connect with the internal contacts in your iPhone.

2.

See these Apple articles below:

If you see No Service, Searching, or SOS on your iPhone or iPad

If you see No Service, Searching, or SOS on your iPhone or iPad - Apple Support

Manually update your carrier settings on your iPhone or iPad

Manually update your carrier settings on your iPhone or iPad - Apple Support

Update your network provider settings manually on your iPhone or iPad

Update your network provider settings manually on your iPhone or iPad – Apple Support (UK)

If you see a Cellular, Ultra Wideband, or Apple Pay Issue message after an iOS update or restart

If you see a Cellular, Ultra Wideband, or Apple Pay Issue message after an iOS update or restart - Apple Support

If you can't make or receive calls on your iPhone

If you can't make or receive calls on your iPhone - Apple Support

3. 

iPhone 6 Plus is > 8 years old, which also affects the battery life. The hardware remains old, so does the battery.

If it is still possible, have the battery replaced.

The battery can be replaced by Apple or an Apple Authorized Service Provider.

See this Apple article: iPhone Battery Repair & Replacement - Apple Support

If you are going to replace the battery, see -> Get your iPhone, iPad, or iPod touch ready for service

Get your iPhone, iPad, or iPod touch ready for service - Apple Support

The price is approx $ 69.

If you want to know exactly, call Apple directly, see Contact Apple for support and service -> Contact Apple for support and service – Apple Support (UK) 

or 

call 1-800-MY-APPLE

4.

You should consider whether it is still worth replacing the battery, with the knowledge that iPhone 6 Plus is not eligible for an update to iOS 13 and higher.

The most apps require iOS 13.4 or later.                                

Considering a newer type of iPhone?

Read this before you buy a newer type iPhone.