Is there a way to "migrate" my current configuration/setup from the old box to the new box?
No there is not. That is a major omission in RouterOS.
It will depend on your actual configuration what is the best way to proceed.
The first thing you can do is use /export file=config to export the configuration to a single readable file, download the config.rsc file and examine it in an editor.
(when you are on RouterOS v7 also include the show-sensitive option with that export)
Now get your new router and perform the following steps:
1. connect it "behind" your RB951, ether1 on your LAN, ether2 to a computer, logon, and do an "upgrade" of RouterOS (system->packages->check for updates).
Also decide which version, v6.xx or v7.xx, you are going to use. There are pros and cons to each of them.
2. after the reboot, do System->Routerboard->Upgrade to upgrade the firmware (BIOS).
3. do a "reset to defaults". that is very important! you need to reset to defaults under the recent RouterOS you are going to use, not the factory-installed one.
4. now it depends on what you want to carry over. You can cut/paste sections of config using your editor and a telnet or ssh session to the new router, but I would advise not to carry over the firewall. Build that from scratch, using your exported config as a guideline what you need.
It is a laborious process, but it is the only way to do it. And it often is a good moment to cut away the crap that accumulated over the years.
Using the cut/paste you can easily migrate more complicated things like VPN or routing setup, or parameters like the internet connection, DNS servers, time servers, etc.
Do NOT just import the whole config as jookraw suggests. That will NOT work.