Swapping your old HDD/SSD to new PC is easy - literally unplug from old machine, plug to new one. Done. Make sure you use the screws to make it stable, especially when dealing with HDD as they're the most delicate part of well, most computers. You don't need different or new cables, old good SATA will work.
Now, was there OS installed on this disk? If so, you'll in most cases need to perform a clean install of your system - or have fun running it in safe mode, dealing with drivers etc. If you new rig have hardware from same manufacturers then maybe it could work without it, but still doing a fresh install would be recommended. In most cases if you try booting different PC from disk that have Windows install from another PC, you'll run into BSOD (Blue Screen of Death).
Without OS? Then no problem, swap and your new PC will recognize it without issues.
As for other questions, if you're on Windows 10 then you should have key. Use it and if activation doesn't succeed I recommend calling Microsoft's support center - they should clear it for you. There's some cases it may not work, but if you had bought OS yourself and not running on OEM system, it should be fine.
Other tips on saving money - well you could tell us your build and somebody may chip in with some potential savings.