How do I ask for signatures in a specific order?

Set hasSigningOrder: true and give each signer a number. Two signers who share a number sign as one group — the next number is not told anything until every signer in the group before it is done:

"signers": {
  "A": { "email": "reviewer@acme.example", "fullName": "First reviewer",
         "order": 1, "fields": [ /* ... */ ] },
  "B": { "email": "approver@acme.example", "fullName": "Final approver",
         "order": 2, "fields": [ /* ... */ ] }
},
"settings": {
  "documentId": "...", "subject": "...",
  "hasSigningOrder": true
}

Want the two of them to sign in parallel instead, as a pair, before a third person? Give both of them "order": 1 and the third person "order": 2. Leaving hasSigningOrder at its default false is the same idea as giving everyone the same number: no order, everybody notified together — and in that case order must not be set at all (it would never be honoured, so the request is refused rather than silently ignored).

Does the whole group have to sign, or just one of them?

Each signer also takes a groupMode, and it means nothing on its own — it describes the group a shared order forms, so every signer sharing a number must agree on it (or leave it unset and let somebody else in the group decide):

Value	Means
"AND" (default)	Everybody sharing this order must sign before the next group is told anything — the behaviour this product always had, and what you get by leaving groupMode out.
"OR"	The first signature in the group is enough. Everybody else still pending in it is marked "skipped" in the response, their link dies immediately, and the next group is told right away. A skipped link opens a page that says the link is no longer valid, and nothing can be done with it; a request already in flight from that page is refused with RECIPIENT_NOT_REQUIRED.

The normal shape — two named approvers, either one will do — is two groups of one signer each, one per group:

"signers": {
  "A": { "email": "finance-a@acme.example", "fullName": "Approver A",
         "order": 1, "groupMode": "OR", "fields": [ /* ... */ ] },
  "B": { "email": "finance-b@acme.example", "fullName": "Approver B",
         "order": 1, "groupMode": "OR", "fields": [ /* ... */ ] }
},
"settings": { "hasSigningOrder": true }

Both share "order": 1 — one group — and both say "OR": whichever of the two signs first satisfies the group, the other's link dies, and there is no second group to wait on. Put the two approvers on different order values instead (still one signer each) and you have two sequential groups, not one OR'd pair — order is what makes a group, groupMode only says how it behaves.

Full settings reference: Signing on its own.