Can I put customers into groups and give each group a discount?

Yes, with the Customer Groups plugin. Go to Customers → Customer Groups:

1. Click + Add group and enter a Group name (e.g. "Wholesale").
2. Optionally tick Free shipping for this group.
3. Under Discount rules, click + Add rule — choose Percentage or Fixed amount, apply it to the Whole order or Specific categories, and set an optional Min cart and Max cap.
4. Save the group.

To add someone to a group, open them under Customers → Manage Customers and pick the Customer Group on their profile. The group discount and free shipping apply automatically at checkout when the customer is signed in.