A few tested routes exist depending on volume and urgency. Whole-household moves go by sea freight, cheap per kilo but slow, or air freight for the fast essential subset. Reputable international movers manage packing, insurance, and customs clearance; compare several written quotes and verify membership in an international movers' association, since this industry attracts fly-by-night operators.
For individual parcels, you trade off among the major couriers, which are fast, trackable, and pricey; national postal systems, which are affordable but slower with patchier tracking on some routes; and consolidation or luggage-shipping services, which suit medium loads like books and clothes. Paying for extra airline baggage during your own trips is often the best value for small precious loads.
Two consistent lessons from expats: insure anything you would mind losing and photograph contents before sealing, and learn the destination's duty-free import threshold and restricted items list from the official customs site, because unexpected duties and stuck parcels cause more grief than lost ones.