Accounts, then contacts, then opportunities and their line items, then activity and campaigns — parents before children, always. Migrate in tranches so each object can be validated before the next depends on it, and keep a reconciliation count per object per tranche.
Use bulk API only where the volume demands it; a change set or a smaller staged load is safer and easier to reverse for configuration and reference data. Reserve the API budget for the objects that genuinely need it rather than burning it on the first pass.
Every claim above comes from work LeanScale published. These are the sources.