How are wallets updated with new balances?
your question is unclear on verification. I think you are asking for this process: You create an address, give it to your friend, he creates a tx, and sends it to you. At this point in time the transaction is not yet mined, but you anyhow get within seconds a message, saying someone sent you some funds.

In the background the wallet software is constantly looking for updates on the network, and here for s.th. like "my address". This eventually triggers the  "funds sent to my address" message on the wallet. At this point in time it is unconfirmed. Roughly after 10minutes it should get its first confirmation, which means it is also included in a block (and verified in all details).

SPV wallets don't do verifications in the same way as full nodes. Full nodes verify each transaction, as described here.