Most practical items are obvious and can be purchased at Target or Wal-Mart, so I'll set that aside.
If you have special essentials, find them quickly. I knew where the health food store was on Day One.
I've had to move many times in my life. And the process of really getting settled and comfortable can take two to three years. Here are some things I suggest.
Find a safe place (neighborhood or park) and take walks. One of the best things I ever did was just wander my own (safe) neighborhood randomly, letting myself get lost then finding my way home. I also walk in gardens (the park and Japanese waterfall in San Antonio, TX, and the Morikami Museum gardens (see my hub on Zen gardens) here in Boca Raton, FL), and on fit-trails.
Find a few favorite restaurants.
Find what YOU like. My wife and I love cooking our own ethnic meals, so ethnic grocery stores were key. These days, the local library is a great resource, also the art movie theater.
Of course, if you're single or more extraverted, you will want to find places to meet people with similar interests. That could be anything from a museum to a singles bar - whatever floats your boat. Wherever you land, Meetup.com will probably help.
As you do all this, don't push yourself and also don't hide. The human animal needs to alternate between exploring and curling up in a familiar lair (book, TV show).
Those are my thoughts - I hope you enjoy your new city quickly!