It is good you are thinking about stuff now! The sooner you start, the easier things will be. I just moved yesterday ;). Start your trash, donate, and pack piles! If you have stuff you would like to sell on craigslist/ebay, START NOW!
Start your mail forwarding a week or so before you move (free in the post office if they actually have the forms, $1 online) and update banks, family, insurance, license, amazon, employer, alma mater, voter registration etc. with the new address.
Take pictures of the new place before you move in for security deposit purposes- can even email them to the new landlord beforehand so you KNOW you are on the same page. Email your current landlord with the date you are moving out and the address you are moving to for legal/security deposit purposes (your state may require a certain number of days as notice).
That is interesting that you still need your own internet in a co-op. Who needs cable? And renter's insurance is usually $10-15 a month from my experience. Definitely worth it.
Have an overnight bag with a set of clothes, toothbrush/paste, scissors/box cutter/toilet paper!!/bottle opener/set of utensils for easy meals :), etc. so you don't have to unpack everything right when you arrive. Make sure you leave out a few rags/broom/cleaner/vacuum so you can properly clear out your old space.
Put everything you possible can in boxes and label them. Don't think you will remember where you put the silverware. Your friends will have a much happier time if they arrive to everything already packed and in nice, closed, packable, stackable boxes. If there are multiple people involved in the move, I color code the boxes with colored painters tape.
Electricity- unplug things when they aren't in use: coffee grinder, hair dryer, microwave, toaster, chargers, etc. Blankets are cheap heaters, keep the fridge full, don't watch TV all day!
Random other things- I use an old set of sheets to cover my mattress when I move so I don't get dirt all over the mattress or the sheets I sleep on. If you have a bed or other furniture that requires assembly, put all of the parts and any tools required to assemble them in a labeled baggy/tupperware. If you are transporting food that needs to get into the fridge, make you sure you put it in a separate bag/box and unpack that first. Stack dishes sideways like records (you can even wrap and stack them in a dish drying rack if you have one and stick it all in a box). Wrap knives in foam/bubble wrap/tape so they don't stab everything and everyone. If you are moving something heavy with drawers (dresser), I've known some people to keep everything in the drawer, remove it, wrap it, and use it as a "box". Easy. Move early in the day. Nobody like packing/unpacking in the dark. Put liquids in baggies so they don't ruin everything else.
Man, I have moving on the brain! The amount of things we sold on craigslist more than covered the amount we spent moving....by a lot!