What are the Best Small Load Moving Companies

3 pros!
I'm moving family mementos like washer, dryer, a few small tables, and some dishes plus chair.
Question: What is the best moving company for small loads across country?
Answer: You have several choices, depending on how much you have to move:
1) Self-pack and ship USPS Parcel Post (for quite small quantities) or UPS/FedEx Ground - you can price shipping costs by individual box size and weights at their websites. Generally limited to certain overall dimension limits and 70 pounds per item. This is what I would do for up to maybe 500# worth, but your furniture and appliances are too big for this. Might consider for your memorabilia (assuming not inherently valuable) using traceable package shipping method.
2) Take to a packaging store and have packaged (or prebox yourself for less cost) and pay to have it palletized and shipped as a pallet. Cheaper but a bit more hassle to do it yourself at a Less-Than-Truckload trucking company - but generally you have to deliver and pickup all boxes as one load so may need a pickup or van or large SUV to carry it all, depending on how much stuff you have to ship. But can pay about $50-100 more (at each end) to have them come and pick up a pallet you have loaded and shrunk-wrapped yourself - they can bring a liftgate truck with pallet handler to load/unload it. Generally cheaper per pound than #1 but more than #2, but commonly somewhat faster. This is the route I would go with not more than a pallet load worth.
3) Go with a commercial mover and have them come and pack and move - you can estimate using estimating calculators at nationwide mover websites, and most will come give you a free estimate or firm moving bid. Small shipments like yours they will commonly crate the chair and small tables in a wooden shipping crate, boxes and washer dryer loose shipped on a moving van moving a number of part shipments - can take several weeks for cross-country move for it to get there as truck makes many stops en route taking on and discharging small shipments. Generally cheaper per pound than; but unless your memorabilia items are in a wood crate maybe not as secure - loose boxes can go astray as shipments are loaded and offloaded. Generally cheapest for larger quantities, not only because they are used to this type of move so do it efficiently, but also your Household Goods moving rates are generally cheaper than miscellaneous cargo rates at LTL truckers who may not have a household good tarriff.
4) Most secure but not least likely to break them - send valuables or memorabilia you want minimum risk for via FedEx second day - though you need someone there at other end to receive it (maybe new employer mailroom will hold small number of boxes for a couple of days ?). Or have a friend hold onto to them till you get there, then ship to you. Certainly most expensive - several to five $/lb typically depending on carrier and which shipping method you use (ground or air, next day or 2nd day, insured or not, etc.
Consider seriously what these items are worth to you if paying for the move yourself - many types of tables and chairs and such, if you are not sentimentally attached to them, you can replace at the other end at a Value Village or Salvation Army or such cheaper than shipping, especially if you have a vehicle to move them to your residence.
Same philosophy on older washer and dryer - if old and not in great shape can be almost as cheap to replace ($300-450 each) as to ship unless doing a moderatealy good sized household move.
Not hard to go to shipper websites and figure out shipping rates per pound (once you get past first couple of pounds, where rates are different) and figure out what method is cheapest for you. For cross-country move with the furniture you are talking about, probably in the ballpark of $1.25-2.00/lb using LTL shipping or a Mover.
And don't forget insurance - your renter's or homeowner's insurance will generally NOT cover in-shipment or in-transit storage losses, and the default insurance will generally (if any) be limited to $0.60/lb - a very small fraction of what you loss would normally be, which is commonly $10/lb or more. For items worth more than $100/lb you need high-value item special insurance, and cannot be sent with some types of shipments.
You can find lots of good info on the web at government and nationwide mover websites - google phrases like - household goods move insurance - or - household goods moving restrictions - and so forth.