Once off the bine, your hop cones need to be dried, as quickly as possible.
Unless you have the time and space to build a hop kiln I recommend a food dehydrator on the lowest setting, usually 95 degrees. Within 24 hours your cones will be ready for packaging. Most kitchen dehydrators can handle 4-8 ounces (dry weight) of cones at a time. However, once your plants reach full maturity, they'll probably be producing more than that in a single harvest, in which case improvisation becomes necessary. To wit:
There we have the very same extension ladder used to pick the hops, laid across a a pair of sawhorses, draped in a bedsheet, and covered in the half hops from one plant while other half is in the food dehydrator. It generally takes about 24 hours for the food dehydrator to work its magic on the freshly picked cones, and 4-8 more for the second batch that has to bide its time on the ladder or whatever jury-rigged contraption you come up with.
At this point, we're all well versed in the importance of freshness. Unless you plan to make a single massively hopped DIPA with your harvest, this is where the importance of packaging and storage comes into play. The basic principles of successful hop storage are oxygen deprivation and low temperatures. Beyond that, it's down to personal preference.
If you have a set of recipes you like to repeat, package your hops per recipe. I'm more of an improvisational sort of guy, so I prefer one ounce packages for mazimum flexibility in recipe creation. It's possible that my long ago collegiate entrepeneurial adventures have influenced this predilection. In any case, it's pretty simple - use a kitchen scale to weigh out an ounce, put it in a freezer bag, squeeze all of the air out, roll it, seal it, and freeze it. Of course, if you have a vacuum sealer, use that instead.
Unless you're single or have a very accommodating spouse, a chest freezer is going to be necessary. Whole leaf hops take up a lot of space rather quickly, much like children's toys, which I swear are reproducing asexually in my living room while I sleep. After the past two weeks of furiously picking, drying, and bagging hops, I have two beer boxes in my freezer filled with them.