Netbackup Tape Management differences from TSM

Switching from one backup product is quite difficult.
I do the same when trying to get to grips with another backup product (e.g. Networker) - I try to map everything back NetBackup.... So, you have all my sympathy!

Firstly - Vault is a separately licensed add-on (unless you have Capacity based license where all agents and options are included).

You need to create different pools in NBU if you want to differentiate between Onsite and Offsite copies in the same way you used to do it in TSM.

Either add tapes manually to these pools, or create a Scratch pool (with Scratch option selected when you create the pool). A Barcode Rule can be added to add all new tapes to this pool by default when robot inventory is done. Whenever a pool needs a new tape, it will pull from Scratch pool. When images expire, tapes will go back to Scratch pool.

You can use Storage Lifecycle Policies to automate duplication to Offsite pool. 
(Without Vault, you will have to eject tapes manually after duplication has completed.)

The SLP is configured for Backup copy and Duplicate copy, specifing Storage Unit, Pool and Retention level for each copy.

You then specify the SLP instead of a Storage Unit in the Policy.

If you need to specify different retention levels for Daily, Weekly, Monthly, etc Schedules, create separate SLPs for each of those.
Create muliple Schedules in the Policy, then override Storage Unit in each schedule and select the relevant SLP.

In NBU 7.6 you can schedule the SLPs.

Please read up in NetBackup Administrator's Guide, Volume I   about media management and SLPs.

Chapter 8 Configuring tape media 

Chapter 9 Inventorying robots   (Barcode Rules are explained here)

Section 4 is all about Storage Lifecycle Policies - Chapters 14 to 19.

PS:

If at all possible - attend a 5-day Instructor-led NetBackup Administration traning course.