As part of my setup process I will put on a new set of D'Addario J74 Medium Gauge Phosphor Bronze strings. For an upgrade charge you can choose from
You may want to consider a mandolin strap. The Mando Shop offers a nice soft leather strap called the Softy Mandolin Strap. It's very comfortable. If you are buying an A style mandolin and want to use a strap, I suggest you have me install a strap button on the underside of the neck near where it joins the body. I find using a second strap button to be more comfortable than connecting the strap to the headstock. The other end attaches to the strap button on the bottom of the mandolin.
I highly recommend Wegen hand-made mandolin picks. They are imported them from the Netherlands and once of the best sounding picks I've ever heard. At a minimum the Wegen 1.4 mm Triangle and the 1.5mm rounded traditional mandolin shaped pick are worth trying as they give you different tone and playability.
The Snark Tuner clips on to the mandolin headstock and tunes based on vibration from the mandolin. It works great in a jam session.
The laminated chord chart is a handy way to have mandolin chords easily accessible. It's durable and rigid enough to lean nicely against something for easy viewing.
A microfiber polish cloth to wipe your mandolin clean before putting it away is the best care you can give the finish of your new mandolin.
The nicely finished wood maple mandolin string winder we recommend that is hand-made specifically for the tuners on a mandolin. The string winder used for a guitar doesn't work well on mandolins because the head of the winder i too big and hits the other tuning keys when being wound on a mandolin. If you plan to change your own strings then this is an essential tool well worth the investment.
If you want to plug in and play through an amp, the K&K Twin Internal mandolin pickup is a very good solution. The pickup is installed on the inside of the mandolin with the 1/4 phone jack replacing the strap button.
All of these accessories can be purchase by clicking on the appropriate picture above.