How was your moving experience? What reliable moving company would you recommend?

Great answer all around. The only exception that I would to counter on would be the part about the "better business bureau" as it is an organization primarily funded by "voluntary" membership dues paid by the businesses that become "BBB-Accredited" and from the BBB corporate partners and sponsors. Yes, I'm splitting hairs here and redirecting a very small part of what was a good comment, but there's very little accountability from the BBB or in the same sense, any of the companies' "rated" when there's such a large conflict of interest from the way that this non-profit is funded. The BBB's "pay to play" model essentially makes it so anything lower than an "A+" rating to simply mean that the BBB won't get paid their "yearly membership dues" or their other various monthly applied fees. The BBB also leans toward the idea of extortion as lower ratings rarely occur until or unless a company stops paying their "voluntary" BBB fees. The BBB, in general, helps to support the Businesses more than they ever do "consumers" meaning actual people. Yes the BBB will take a complaint and yes they may talk to the business for you about your concerns, but don't expect any more than that as the BBB rarely does anything actionable on your behalf. The BBB primarily functions as a franchised business directory or a glorified "yellow pages" and little more than that and anything more for the public is an illusion. So yes, the BBB may be a good place to find moving companies but the BBB rating shouldn't play any part in your decision process of choosing any moving company other than getting their names and contact info.

As for the actual moving process, no matter if you choose to "pack & box everything yourself" or the "sit back and we'll take care of everything" route with a moving company, the only person who's actually accountable during the move is YOU.

Keep in mind what the movers are going to do whatever they will or want to do with your possessions when they move them and how your possessions will be stored. Some major movers can simply drop the larger trailers (like with Mayflower) or loadable shipping container type boxes at a generally close storage location. However this is only possible if they have the availability of tow-able trailers &/or container storage boxes available for that to happen, but never expect that to be the case as many major movers may/and will unload your possessions into a general storehouse facility. Many of these storage facilities are practically wide open once inside where the only delineation between your possessions and those of other being stored are painted lines on the floor surface, small temporary barriers, or partial fencing. With whatever the method you may choose, there's no real guarantee that all of your possessions will make it to their final destination. Items can always be lost, stolen, or damaged either by the movers, the moving processes, or any other number of unforeseen circumstances.

So, as always, get what you want and need in writing beforehand from the moving company you choose. If you have a lot of things then insure your move. You can do that through the moving company or by other means. If you have home or renters insurance, check to see if your possessions & your move is or can be covered by/through those insurance providers too before you start your actual move which may require you to keep your current coverage until after you move is completed, you've unpacked, and have settled in to you new residence. Report any damage, breakage, or loss as soon as possible.

The best bits of advice that I can give after a lifetime of relocating (both as a military brat, and as a member of the military myself) is to do your due diligence and to take an active part in your moving process by keeping a record of your items both large and small before, during and after your move. I'm not saying to make a list of everything that you have and own, but doing that never hurts in the long run as everyone should make and have a home inventory so that you can file a claim (be it from a moving, a renter's, or homeowner's claim) if/when any loss might occur. Unless your move is over seas and out of CONUS, the list you make when moving should be a little less demanding than a full and total inventory of your possessions. You should keep track of the larger major items, like furniture, vehicles, TVs, or computer like devices that you can't physically take with you. And as a general rule you should also count and keep track of the total number of boxes during the move so that the number of boxes, both going out during your move and the number of boxes coming in at your new location are the same. They won't like it, but don't be afraid stop the movers so that you can do your counts, or if you see any problems in the way that they handle or move your belongings. Be polite but be firm and clear in what you want from them, but please don't try to micro manage any of them and just let them work until or unless an interruption is necessary or a simple act of kindness of giving them something they need. Whether those counts you do are for your entire home or room by room is up to you, but keep a record as it will only be a benefit in the end.
Also, providing some food and hydration like the CWO3 said in another comment is always a good idea too. Maybe throw in a place to sit in/at or depending on the temps & weather, maybe an area that they can cool off or warm up in when needed.

Good luck in your relocation and/or change of duty station.