I bought a domain somewhere else and tried changing the A records there to the DNS nameservers that Dreamhost uses (22.214.171.124), but it didn't work. Is there any way to have my domain redirect to Dreamhost (I have a hosting account there) by changing only the A Record? If so, what IP address should I use?
You might need to brush up on your DNS knowledge a bit but regardless there are two ways to point your domain name to a server:
If you're DNS registrar (you) will provide a name server to answer queries about your domain then you need to setup NS records. NS records for a domain tell the client who to ask for information about the domain. Then on your (or your registrar's) name server you would simply point a new A record to the static IP address provided by Dreamhost.
If you must use Dreamhost's DNS servers (fairly common with shared hosting) you simply need to point your domain name (through your registrar's system) to Dreahmosts DNS system. They provide pretty detailed documentation about how to do all of this on their wiki.