Depends on what you mean by best? Do you mean fastest, cheapest, or safest? Overnight at the USPS is the fastest but god awful expensive.
UPS is pretty fast, cheaper but riskier because there is no insurance protection in case of loss or damage (Neither UPS or FedEx insures coins, including collectible coins. It's in their terms. The best you could expect to et back is the insurance fee that they collected.)
First class insured is fairly quick, but the package travels with the regular mail and the insurance sticker is a big ol flag that says "I'm valuable, steal me!" Fortunately the over $250 insured does get scanned as it goes along so you have tracking along the way. The under $250 is just scanned when it is accepted and when it is delivered so they have no idea where it is or where it might have gone missing it it does disappear.
Registered is the absolute safest way to ship valuables. They are kept under lock and key at night and everytime they change hands they have to be signed for so there is always a specific person who is responsible for the package every step of the way. If one goes missing the last person to sign for it is responsible for it and it can cost him the value of the item, jail time and at least his job and probably pension. Because of this safety you can insure Registered mail for higher amounts. First class mail has a maximum of $5K, Registered has a Maximum of $25K. Also do to the additional safety the insurance rates are lower Once the package value exceeds about $800 Registered mail is actually cheaper than first class insured and as the value goes up it becomes a LOT cheaper. (Once you get to a certain point Insured mail rates increase at $1 per hundred insured value, registered rates increase at $1 per thousand in insured value.) The big downside to Registered is that it is SLOW. What may arrive in three or four days by regular mail may take as much as two weeks or more by Registered. But you can be pretty much absolutely sure that it WILL get there.