> Is that possible for mobile provider to read our chats and sms?

Yes for SMS/MMS. The provider always has access to the message once it
reaches their network.

Chat depend on the chat program you are using. Some are better than others.

Chat also depends on the other party. Chelsea Manning (née Bradley
Manning) used a secure chat with OTR messaging. His peer, Adrian Lamo,
threw him under the bus. Lamo kept transcripts and turned them over to
authorities even though Manning thought his message were off the
record (OTR).

> How to prevent them from being read?

You don't. You switch technologies because technologies like email and
SMS/MMS are impossible to secure.

> I have installed a firewall that creates VPN on my Android phone, but it 
> doesn't seem to close all privacy holes.

The firewall will stop some network threats. But there are other threats.

VPN is usually used for confidentiality, and at a layer lower than
SSL/TLS. There are some things that make it better than TLS (for some
definition of "better"). But its usually harder to configure.

> They still can read what i wrote.

It depends on the application and protocol.

No one is likely reading your traffic between the the device and your
home server if you are using a VPN.

> How to entirely guard our network connection from prying eyes?

Encrypt all the traffic you can. Don't use services that don't provide
end-to-end encryption.

(End-to-end is the key; and its often where TLS falls on its face).

Jeff

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