You have mainly three options after setting up you new core server:
1. Install an agent from your new core manually
2. Schedule a push distribution from your new core server to all clients
3. Schedule an advance agent push to all your clients from the new core server
All these three options will move all your clients to the new core server at the same time as update them to service pack 2
If you would like to keep your old core, you can install SP2 on your core and could push the sp2 client update via Patch Manager to your clients.
Nevertheless, you need to reinstall your agent or just push the sp2 to your clients, depending if you want to keep your old core or move to a new one.
No, new agents or agent setting updates need to be distributed.
You can make a column view with the agent version in it.
@Casper: How do you want to update your agents to SP2 if you don't distibute a new Agent? Or at least the Service Pack trough Patch Manager?
Maybe I missed something, but I thought the question is, what needs to be done to move the clients to the new core server AND update the agent to SP2?