I've been in the moving business for the past 12 years, starting as a
"lumper" and as a driver for the past 7 years. Go with a big name company
(United, Mayflower, Graebel, etc.), they won't pull the "fly-by-night" scams
you hear about.
If you can help it, don't move during the peak season (May-sept.). That's
the time when ALL the companies are so busy they need to use temp labor with
no experience. Also, don't go for the biggest discount. A good driver will
turn down the load if it doesn't pay enough, so you'll be stuck with a
second rate driver. Specifically as for a low claims driver. Also, pay the
little extra for "Full Replacement Value" on your shipment. That way if
anything happens, the van line/driver is on the hook to repair or replace
any damage item. Otherwise all they're liable for is $.60/lb.
Generally, any reputable, full service moving company should be able to
handle your machinery. We regularly move grand pianos, gun safes, and all
manner of machinery without any problems.

Mark