Not to start. An ordinary printer and tape will carry you a long way, and plenty of stores never buy one.
What there is to print
A packing slip, which Metzee generates for you — a proper parcel slip with a large ship-to block and your store on it. You can print them one at a time or in a batch from your orders.
Per-order slips leave the prices off, so a parcel can travel as a gift without an invoice in it. That is worth knowing before you handwrite anything.
The courier's label, where the courier produces one.
The cheap way, which works
Print on ordinary A4, cut it out, and tape it on — covering the whole label with clear tape so rain cannot touch it. Thousands of parcels go out like this every day.
The failure mode is tape over a barcode making it unreadable at an angle. Use clear tape, lay it flat, and do not stretch it.
When a thermal label printer becomes worth it
Volume. Somewhere around a dozen parcels a day, the minutes start to matter. Below that it is a gadget.
Running cost. Thermal printers use no ink, which is the real saving over time — ink is what makes an ordinary printer expensive per label.
Durability. A thermal label does not smudge.
If you are packing in batches already and the printing is the slow part, it pays for itself quickly. If your bottleneck is finding the item and wrapping it, a printer fixes nothing.
What actually speeds up packing
In order of effect:
Printing in a batch rather than per order.
One box size for most of your range, so there is no decision.
Everything in reach — tape, filler, scale, slips — in one place.
Weighing as you go, and writing it on the box. What is chargeable weight?
A printer is fourth on that list at best.
If you have no printer at all
Write the address by hand, clearly, in capitals, and tape over it. Entirely acceptable.
Keep the order number on the parcel somehow — it is how a problem gets traced.
Many courier counters will print from a booking on their system.
More: how to pack an order, what packaging should I buy, how to mark an order shipped.