I have a Dell Dimension 3000 desktop computer with the basic 256K memory. I have ordered a 1Gig upgrade memory card from Dell. In order to get maximum performance do I remove the original 256K card and put the new
1Gig card in that slot or leave the original in place and install the new card in the remaining slot?

With memory, more is generally better. So, yes, you can leave the original in place and install the new module in the second slot, however, the modules work best if they are matched for size. See Removing and Installing Parts below for a discussion about installing mis-matched modules.

Also, if the modules are capable of different speeds with the new one capable of being operated faster, installing both will cause both to operate at the slowest speed. Depending on how the computer is used, faster may yield better performance than an extra 256 MB of quantity, in which case installing only the larger module would be better. The actual speed is dependent on the installed processor and the front side bus speed. See the Technical Specifications below.

So, to determine the configuration that yields maximum performance with mismatched modules for size and/or speed, benchmark testing would probably be needed. Yet, with 1 GB DDR400 modules running $30-40 online at places like Newegg.com, consider maxing it out at 2 GB by purchasing another 1 GB module that matches the one already purchased for size and speed, and installing a matched pair.

Resources:
Dell™ Dimension™ 3000 Systems Service Manual - Technical Specifications
Dell™ Dimension™ 3000 Systems Service Manual - Removing and Installing Parts
Kingston Technology
Newegg.com