Seed beads are close to my heart, they're one of the first crafty things I ever got into. They're incredibly versatile, and in this case pretty festive, too. They work well with math, a lot of common beadwork stitches are influenced by fractal-like increases and, in this case, I'm working with the relationship between hexagons and circles. (Is there a special term for that relationship mathies? I couldn't find anything easily with google!)
Don't let the tiny beads intimidate you. Start with small eggs and work up to larger projects. I make up my patterns as I go, but you could also make up a chart and plan that in advance.
Step 1: Supplies and Equipment
Eggs - wood are nice because they're sturdy. Base coat them to match your beads. Plastic would work, real are kind of risky (though pysanky people use real eggs, so it works for them!)
Seed Beads - these are size 11, size 15 is smaller and therefore would be better but more tedious.
Beading Thread - this is nymo in size OO. Use a beading thread, not a sewing thread.
Beading Needles - I use size 12 needles.
Scissors - tiny points are helpful for trimming thread.
Step 2: Peyote Stitch
One of the classic beading stitches is called peyote. It works up quickly (for beadwork.) You add a bead to every other column in each row, and the beads end up in a brick style layout. It is well suited for diagonal and vertical lines, horizontal lines not so much. If you've never used this stitch before google it and make a few things with a flat, even number of beads peyote.
Step 3: Measure the Egg
String seed beads onto a piece of thread. You're trying to measure the widest point on the egg, the "waistline" if you will. Keep stringing beads until you have the highest multiple of 5 without the strand being longer the the waistline. This is the length of the longest row. Make sure to write down and/or remember this number.