By the Tenant
If you want to move, there are several steps you can take to protect your credit rating and protect your ability to move at your own pace.
Look for a new place to live in advance. Try not to give notice at your old place until you are certain of your new house/apartment.
Give the landlord written notice you are moving. Generally, in a month-to-month tenancy you have to give 30 day's notice to the landlord that you intend to move. For a week-to-week tenancy you need to give 7 days notice.
Remember that you cannot legally use your security deposit as last month's rent. If you give notice June 1st that you will be moving July 1st, you are still legally obligated to pay June's rent. If the landlord gives permission to use the security deposit, get that permission in writing. Just write on a piece of paper "Larry Landlord permits Terry Tenant to use the $800 security deposit as the rent for June.". Date it, sign it, keep a copy.
Try to leave the apartment fairly clean. If you leave a lot of junk in the apartment and the landlord has to hire someone to take it away, s/he may not give you back all your deposit or may sue you in small claims court for the cost.
Take pictures of each room. Use a camera that puts the date on the photo or take a picture of a newspaper in with the roll of film to prove what date you took the picture. That way, if the landlord comes after you later in a suit for damaging the apartment, you will have proof of the condition of each room when you left.
If you want to get your security deposit back, make sure you give the landlord an address where s/he can send the deposit and/or the accounting.
If you are receiving Section 8 assistance, you need to notify your worker and request a new voucher. You will not get another certificate since all certificate are being changed to vouchers. Ask your worker what this will mean to you since your rent might go up. Many times, if you want to move, the Section 8 worker tells you to give a "30-day notice of intent to move" to the landlord. This is dangerous since, if you cannot find a new place and get it inspected and move in that time, the landlord can try to evict you based on your own 30-day notice. Or, you may not get the new voucher, and may be stuck in an eviction action based on your own notice. Speak with an attorney who knows about Public and Subsidized housing. The attorney may be able to negotiate with your landlord which protects your ability to stay in the unit if the new voucher falls through. Get a letter from your worker that you are definitely going to get a new voucher.