Disclaimer: Do your own research! This information worked for me, but it may not be suitable for you. I am not responsible for anything that goes wrong when using this information.
This information applies to the following residential furnaces:
Carrier 58MVB, 58MVC, or 58MVP
Bryant: 355AAV, 355BAV, 355CAV, 355MAV
Payne: PG9UAA
All part numbers in the description below are current as of the date of this post. Manufacturers have a tendency to update part numbers, so do your research. The Carrier Enterprise site seems to be a good source of information, but unless you are an approved vendor you only have access to a minimal amount of information.
If your furnace is making a lot of grinding noises, your draft inducer assembly is probably going bad and needs to be replaced. This is an expensive part, but it is a small fraction of the cost of a new furnace and the replacement is something a reasonably handy homeowner can do on their own. The current draft inducer part number is 340793-762.
You will also need to ensure you have the correct pressure switch. If you have an old draft inducer assembly, your pressure switch will not fit on the new assembly and you will need to get a new one. If you have a 40,000 BTU/Hr furnace, you are going to need part number HK06NB025. For all other furnaces, you will need part number HK06NB023.
These pressure switches can be mounted to the bracket that comes with the 340793-762 draft inducer assembly, but you will need to find a few screws to make it happen. There are plenty of mount points on the pressure switch bodies, just be sure your screws are not so long that they pierce the plastic body of the pressure switch.
While undertaking this repair, it is important to check that the condensate trap and the hoses leading in to it are draining reliably. If any of these are plugged, condensation will back up and damage internal components of your furnace.
You might also find that some of your hoses are in bad shape. Part number 319873-715 is the hose that connects to the pressure switch (it has a pink label), and part number 319873-714 (green label) is the hose that connects to the condensate trap.
You should be able to find maintenance, troubleshooting, and installation manuals online. They will help a great deal when undertaking a repair like this.