How to make sure luggage makes it across multiple airlines?

Simply put, the best way to avoid problems with checked luggage is to never check luggage. Reduce the things you carry ruthlessly until they can fit into a carry-on bag, or consider a shipping service— FedEx or DHL will happily send it straight to your hotel, and you'll be able to insure it and track its progress online. In fact, I would prefer to ship some very important items, for example, a wedding dress on its way to a destination wedding.

Of course, the "no check" option is not always feasible, especially for extended trips; there are too many things needed and it is too costly or time-consuming to ship. And non-stops are not always available or cost-effective.

Check baggage loss statistics for airlines, although these are not always available or up-to-date. For example, the U.S. Department of Transportation publishes Air Consumer Travel Reports on a monthly basis, with mishandled baggage being one of the metrics covered. Of course, your mileage may vary; large international network carriers will always have more mishandled bags than domestic point-to-point carriers, and you won't be able to take the latter on an intercontinental trip. I would say airports, rather than airlines, are more closely correlated to lost bags, but those stats are almost impossible to come by.

There are devices and services such as Trakdot that will alert you if your luggage does not arrive. You might look into RFID tracking services like ReboundTag or SuperSmartTag. These won't prevent your bag from getting lost, but makes it easier for the airline to track them down and match them to you, making a faster recovery possible.

But would not be in your best interest to book each segment separately. You don't actually reduce the risk of it getting lost: a bag on one leg can fall off the truck and never get loaded, get knocked off the conveyor belt at the terminal and never make it to baggage claim, or get stolen by an airport employee just the same as a bag scheduled for multiple legs.

Meanwhile, you swap the potential inconvenience of losing a bag (and probably only temporarily) for the guaranteed inconvenience of needing to reclaim it, process through customs, then re-check the bag at each transfer point. On separate tickets, if your inbound flight is delayed or canceled, your connecting airline has no obligation at all to you to accommodate you on a later flight; you would be subject at the very least to a change fee and fare difference, and if you are not able to cancel the ticket prior to its scheduled departure, you may forfeit its value entirely and be forced to buy a walk-up fare or to cancel your trip.

As a rule of thumb, you can't guarantee safe delivery. There is always the "possibility"!

There are specialized services for Door to door baggage transport. A commercial option available online is one that British Airways (read never cheap) recommends is First Luggage. You could look for similar services offered by other companies.

In the same line of thought, you could always get a logistics/courier company to haul your baggage for you separate from the airlines.

Also, ensure that you split the essentials across the multiple pieces of luggage, as when the inevitable happens, you are not left with only souvenirs to cover your essentials!

As you have made no mention of any budget constraints in your post, I assume there are none to a reasonable extent.

The only way to ensure that your luggage does not get lost is by carrying it on. I'm on the road 25 weeks a year and almost always do carry on - there's usually a way to trim back.

The few times I have checked my luggage, I would estimate it has been lost 50% of the time. The good news is it's extremely rare to lose luggage (they assure me) and less then 1% of "misdirected" bags are lost permanently. Usually it misses a transfer and arrives on the next flight. You get it in the morning, and even drive it out to your hotel and leave it with reception. If it's heavy you sometimes hope it's their problem.

I wouldn't worry about it though. If you can't restrict to carry on, put everything you need for a couple days and you expensive things in your carry on, and the rest in your checked. Have fun!