Right now I have a 3TB WD red HDDs. It is old, almost 10 years old plus I am running out of space.

I know I can add drives but I will do a complete pool reinstall. I will backup my entire file system with a 4TB external HDD.

Can/should I, get 2 smaller HDDs (6-8TB) or a large (20-22TB) HDD? Its been a long time since I setup TrueNAS so I do not remember which way to go.

You want multiple drives of the said size for raidz2.

Yup, 3 drives minimum for z1 for parity.  
zfs calc

Unraid doesn't have any of these limitations. You get a single (or two which is just the single mirrored) parity disk that covers all drives. You can add any size data drives (as long as they are not bigger than the parity drive) at any time and it will increase the size of the storage pool. You can also pull any drive and they are data readable in any machine. Its perfect for home use large file/media storage. The only down side is speed compared to traditional raid.  If you need speed for many quick transactions look elsewhere.

https://unraid.net/product

Well I bought one 10TB drive. I guess I will be buying 2 more.

This is what you need^  https://diskprices.com/  for price sorting.

I know have 3 x 10TB HDDs. So, raidz1 or raidz2? In the future I will add an all SSD backup server to backup my NAS.

SSDs as backup wouldn't be my choice, but it's your money.

Check that calculator I linked above to see the difference you have with Z1 vs Z2.  AFAIK 1 vs 2 parity stripes and in bigger arrays Z2 can tolerate more disk issues.