My tried and true packaging system -
Item in appropriate sized box with padding as needed. If it can spill at all or is fragile - wrap in paper towels, fabric scraps or similar, seal in a Ziploc, wrap in bubble wrap, put in a large Ziploc, then pad box. Less likely to break, but also won't make a terrifying mess if something does break.
Box bottom - over seam, overlap two pieces at least per seam side. You want at least half the bottom covered. Works far better than trying to tape "ends.
Put your boxed item in a bigger box. Add packing material as needed.
For very delicate items - egg crate mattress pad. Cut to line the box. Also, if small enough, cut styrofoam box in half, dig out inside to nest item in, tape around cut seam, then around and over the block.
Glue or tape address on inside and outside of both boxes. All should have water resistant covering of some sort.
Do not prewrap presents. Include the material, but wrapping may not get through scanning intact.
Seal the top same as the bottom.
New, unmarked boxes are sturdier and less likely to get too lost in the mayhem known as Christmas shipping. Outside of heavy shipping times, a sturdy, clean box with minimal writing that's fully blacked out or covered is good enough.
Insurance and required signature aren't always needed, but now is when things get lost due to high volume and thief goes up due to high number of packages.
If you are shipping international - be very aware of any custom issues. Kinder eggs may be delightful, but the potential fine per is $2,500USD and you don't get the egg. Various chemicals may be legal shipping between states, but not across borders.
USPS remains the most reliable for normal shipping, for next day or second day, Fedex wins, but you will pay for the service. At my work, standard is $10, second day is $20, next day is $55, and next day 10am is $105 - all are to start, NDA and NDA10 frequently go over that.
I worked in packaging far too long, in case you can't tell. But, I've frequently shipped international for personal mail and never had a lost package. Slow, yes. Damaged packages, yes. But, it's always gotten there.