Hi there,
It really depends on your components and what the old and new cases are - but assuming the boards and PSUs are all ATX standard then you shouldn't have much trouble.
This is the order I would do it in:
- Turn the PSU off at the wall at the unit
- Disconnect all the power connections from the PSU to the components
- Disconnect all the data connections and put them to one side
- Unscrew the graphics card retention screws and disconnect it from the motherboard (remember there is usually a press-down catch on the PCIE slot) and put it safely aside
- Unscrew the PSU and put it to one side
- Unscrew the motherboard (with the CPU cooler and RAM sticks still attached) and put it on an non-conductive surface (cardboard or wood is good)
- Disconnect all the other things you are transferring to the new case like fans, HDDs, DVD drives etc.
- Get your new case ready to accept all your components (attach fans, remove packaging, pop out drive bay covers etc.)
- Connect up your DVD drive, fans and HDD/SSDs to your new case
- Depending on the design of your new case put the PSU or motherboard next
- Install the graphics card
- Connect up all the power connections (including case fans)
- Connect up all the data connections (including the front panel connections)
- Plug the PSU into the mains and turn on the PSU
- Test the system
Also, while going through the build try to achieve good cable management - as it makes upgrades/maintainence so much easier and it will help with airflow a bit.