Depending on what OS you are running on the old system might help determine the best way for people to answer you. If your old computer is a really old outdated system, then it might be better for you to copy your files to an external drive where you can retrieve them manually afterwards.
An iMac with a 1 gig hard drive has to be ancient. I assume the new one is a 256 Gig SSD drive.
Otherwise, give a bit more detail on what system you are running on the old computer and connections capabilities from old to new. If you are going to use Setup Assistant or Migration Assistant, you will either need to connect the computers together by cable or network, or clone the old system to an external drive which can be accessed by the new computer.