First of all it is assumed that generic NON-OEM PCs are involved here in terms of both the "old" PC and the "new" PC. If you're dealing with one or both OEM machines what I indicate here is NOT applicable.
That being the case...
Go right ahead and install your "old" HDD or SSD in the new PC. If you prefer to clone that disk and use the cloned disk for installation in the new PC then use the clone for that purpose. Ideally it should be the only drive installed in the new PC at this time. Presumably you'll be connecting the drive to the motherboard's first SATA connector usually designated SATA 0 or SATA 1.
(BTW, your tentative decision to clone your drive and use the cloned drive as the transferred disk is a wise one. ALWAYS, when possible, use a cloned drive when attempting major configuration changes either in a current or different system.)
Assuming that the disk you plan to install in the new PC is non-defective and contains a viable OS there's a strong likelihood that the system will boot to that drive and the data on that drive will be accessible. Is it 100% reliable? Not by a long shot. BUT IT'S WORTH A TRY. (I'm assuming, of course, you're working with a "modern" Windows OS, e.g., Win 7, 8, or 8.1).
Generally speaking Windows is very good at picking up the drivers the system needs, however, it may be necessary to install this or that additional driver the new system needs from your new PC motherboard's installation CD/DVD.
By & by there probably will be a need to activate the new system so have your OS product ID handy.