On new Mac, upgrade old Snow Leopard install or start with clean install? I've got an old 2008 Macbook which is not eligible for upgrade to anything supported at the moment. So, I am looking to buy a new MacBook Pro.

My question is,

Is it better to copy/move my HDD to the new machine and then do an upgrade or start with the clean install of OSX already on the machine and slowly reinstall all the apps I need, as needed?

Thanks.
You're five major system updates back from  El Capitan, and Apple has made a lot of changes to code, file arrangements, database structures, network connectivity etc. in the meantime.
Apple usually does a pretty good job of clearing away the old to make way for the new, but things do slip through. I recently killed a few prefs files from 2007 that Migration Assistant kindly transferred to my 2014 mini. Invisible files and such often gets copied too. iTunes data structures have changed significantly, with loss of artwork or metadata upon upgrade being fairly common.

Apple is pretty good, but they're not infallible. With a jump of that many years, I'd go for the clean install, followed by thoughtful installation of needed Apps, and manual transfer of docs, pics, music etc. That way you'll end up with a thoroughly modern System rather than a System that also contains a hodgepodge of used or unused pieces dating from an earlier decade.