Before I start this process, can I pick and choose files and folders in System and network?

TuckerdogAVL wrote:

can I pick and choose files and folders in System and network?

No.

What you need to do is purchase one or more external hard drives. Use one as an archive and use one for Time Machine. They should both be at least 2 TB. Make a Time Machine backup of the old computer. Then manually archive any large files that you want to keep for posterity. Time Machine is a great backup system, but it is only a backup system. If you delete files, they will eventually be deleted from Time Machine too.

Once you've archived your old files and make a backup, then you can prune the files you don't want to transfer to the new computer. Once you get it down to 300 GB (and not one byte more), you can do the transfer.

Another option would be an iCloud subscription. Make sure all of those files are in an iCloud location and upload them all. Then just turn on iCloud on the new computer. You're done. The Time Machine backup and archive would still be a good idea, however.

Do "System and Network" really comprise that much space? They ought not to. I don't advocate omitting them, but you can if need be.

Create a Time Machine backup as etresoft wrote. Ideally, two or more backups. That gives you options, including the ability to migrate from the TM backup without having to tie up your existing Mac.

Move drive hogging data like movie libraries and music libraries to a separate external drive and keep them there and do not migrate them to the new machine. Music and TV libraries can easily be linked to an external drive.

Personally, I keep all my user data on an external drive and only keep apps and those items that some apps really need on the internal drive. Of course, I have a second drive to back up that one.