give me the types of methods to connect vpc via peering ?

The two methods to connect VPC via Peering,
1. Inter region VPC (same or different account)
VPC peering between two VPC in different region in same or different aws account
2. Intra region VPC(same or different account)
VPC peering between two VPC in same region in same or different aws account.

why we use private ip to connect the another regions vpc?

When connecting VPCs in different regions, we use private IPs because the traffic stays on the cloud provider's secure and high-speed global network backbone, never traversing the public internet.

why don't we connect 2 vpc through internet?

Connecting two VPCs over the public internet is not a good idea because it introduces significant security vulnerabilities, performance issues, and higher costs.

how to connect routing for vpc peering?

To connect routing for VPC peering, you must manually update the route tables for the subnets in both VPCs.

what is the purpose of vpc peering?

The main purpose of Peering is to enable secure and efficient resource sharing between different VPCs