What is the easiest way get from from Shenzhen airport to Lockhart Road, Wanchai?

In the airport, there are direct buses going to Wan Chai area, and the bus fare is CNY130. Upon arrival, you can reach the Lockhart Road within a few mins walking.

And how does it work with the immigration? It's my first time to travel to HK like this, no idea how it works.

If it is an international arrival in the airport, you need to go through the following procedures: Arrive at the terminal building - Health quarantine and frontier inspection - Baggage claim - Customs inspection - Leaving the airport.

When taking the cross border buses, you need to pass the immigration in the Shenzhenwan Port. You need to get off the bus and go through the immigration procedure. Then still take the same bus to the destination.