My concern would be how well the data is locked down by Apple's proprietary HD formatting methods.
Maybe the easiest trick would just to put the old hard drive in the new computer tho there is risk of the hard drive being damaged by the past computer, and that this hard drive could damage the new computer.
Your Sata to Usb idea is perfectly fine for windows computer, not sure if Apple would have driver support for that, but no reason to think they shouldn't, so you may want to buy one to try it out.
I suggest searching for any guides about expanding your Mac's storage abiliety. If the tricks used for such guides (such as a simple exsternal hard drive using the SATA to USB thing) work, then there is no reason to think such tricks won't work for your situation as well.
If you actually want to transfer the data to the new hard drive, and the SATA to USB doesn't work, you are going to need another computer that supports multiple HD and allows you to swap data around. Mac computers are expensive, and the most expensive actually allow for this kind of support, so it is unlikely that you may have one near by. windows computers can do this because their builds aren't as refined as macs, but then it becomes an issue if the windows computer can handle the movement and handling of apple formated data.