It's easy go to movingcollege.org and use the information they have.
The first thing to do is to determine the date you are leaving where you are and when you will arrive at where you are going. You need to give 30 days notice to your landlord (at least) and arrange for a start date on your new place. Be sure to be realistic and call your new manager if you will be late arriving into town.
Second, call all the utility companies and arrange for turn off in one place and turn on in the other. It is very wise to keep the account numbers handy--use the most recent bills for your files. Obtain estimates of fees that will be charged and put aside the money for them.
Set up your forwarding address information with the post office, creditors, family and so on.
Gather boxes--many stores will hold them if you call ahead. Or, buy some used ones from moving companies. If you can settle on one size that you can carry, you will be more efficient in loading the boxes. Informal packing is fine for in-town moves; but for longer trips, the better packing is worthwhile. If you have books, it usually takes 1.5 boxes per shelf.
For fun: go to a furniture store site and use their online software to decide where everything will be placed in your new home. If it won't fit, sell it or give it away before you move it.
Finally, absolutely definitely and without fail, arrange a walkthrough with your landlord before turning in the keys. This will prevent a surprise bill coming from them for dings and nicks.