When you recover your gamertag, it will recover all of your achievements since those are bound to your gamertag. However, you will have to buy another storage device (such as a USB drive, see steps below), or a transfer cable to move all of your old saves/other data to the new hard drive.
USB device
- Copy all data such as saves to the USB device through the Xbox 360 system / data settings.
- Insert your new hard drive, connect your USB device, and copy your data/saves from the USB device to the new hard drive.