I send everything with Royal Mail. The postage cost is calculated on the size and weight.
All framed art will be sent 'Signed For' as we both need proof of delivery and also insurance should anything get damaged or lost. In the hundreds I've sent, I've only had one damaged and one lost, but it can happen.
I use biodegradable bubble wrap and new or reused cardboard. Occasionally I will use leftover packing foam to protect an item.
I always prioritise the protection of the item over beautiful wrapping, so there may be layers of bubble and card and it may not be very glamourous!