Personally, it's a huge disappointment for me to receive mail – only to find out that it is for a prior resident. Living in an apartment near TCC, there is a lot of coming and going between residents annually, so this is a common problem.
When you move to your TCC apartment, there is an easy way to make sure that you are getting mail from your old address, stopping the problem for that future resident. You can visit the USPS website and fill out a change of address for there. Doing this costs $1. You can choose the time to begin forwarding your mail, so it's fine to do in advance. (Please note: many websites are out there that claim to be official websites to change your address. If it is not the usps.com website, it is not the place you want to be.) Alternatively, you can go in to your local post office and pick up a change of address form there. They will answer any questions you have, and this option is completely free! The USPS website also has some general information on changing your address, which can be very helpful.
This service will forward "important" mail to you in your TCC apartment. It will not forward advertisements or catalogs. Magazines will only be forwarded for 60 days, so you should contact any magazines that you receive and let them know when you are moving. If you know some bills will continue after you move (such as a cell phone bill), you should contact the company yourself, as you will not get forwarded mail as quickly as you would get mail addressed to your correct new address.
In your new TCC apartment, expect to be getting some wrong mail for a little bit. For each piece of mail that your receive addressed to someone else, simply write ‘Return to Sender" on the front of the envelope and stick it back in the outgoing mail. This will let that sender know that the resident no longer lives there.
If the mail persists for months, the post office may not be aware that the prior renter has moved. You can, in addition to returning mail to the sender, write "No longer at this address." The post office may take notice of this note. You can also tape a note outside or inside your box (depending on the practicality based on your mailbox type) that states either "PRIOR RESIDENT NAME no longer lives here" or "YOUR NAME(s) mail only." This may help the mail carrier to remember not to put any mail that is not for you in your box.
As a reminder, it is important to remember not to open or throw away someone else's mail – it is illegal. Let the post office handle it. However, for magazines or coupons and fliers, the post office wouldn't go to the trouble of forwarding it anyway, so I personally do not feel bad throwing these things away.