When you travel to an other area for just a month you prepare as if for a vacation, not for a move.
You do not need to deal with registering in the new town, no change of address, no new driving license and so on.
What you need:
Money and a way to get more. Contact your bank to be sure that your bank card will work where you need to go and at no extra cost.
If you have a credit card, bring it.
ID, state ID or driving license will do, passport if you have it.
Insurance, travel insurance will do.
Check out whether you will easily find a place to stay or whether you need to make reservations.
For a month a motel with a studio or apartment will do. A kitchenette is going to safe you a lot of money on eating out.
You will not rent a room in a shared house for just one month but you might go for a hotel room or even a bet in a hostel instead.
Plan your journey there, car, train, bus, flight, which one depends on how far you need to go and whether you have a car and will need on once there.
Make a list of clothing you would want to take with you for a month. Take clothing that is a bit cooler and a bit warmer than you would think to need, that will allow you do adjust if the weather is unexpected.
Also make a list of everything else you think you need, from phone and address list to couple of photos to remind yourself from home.
If you travel by car, restrict to what you can easily fit in the car, if you travel any other way, restrict to one big bag or case and one small one. Like the airline check-in and carry on luggage.