Do you have iTunes installed on the laptop?
You can plug the phone into said laptop and sync the iPhone to iTunes. This will bring all that info over and store in iTunes.
If it is just the pics, when you plug the phone in the computer should recognize it as a device and allow you to transfer the pics like the iPhone is a camera.
You can transfer files from the iPhone to the laptop. Launch iTunes on the PC. From the menu bar top of the screen click Store > Authorize This Computer Connect the iPhone. From the menu bar again, click File > Transfer Purchases From ... You can do the same procedure on the new Mac and you will have to "authorize" that computer also.