Q: What is recommended in order to speed up an old iMac computer?

How much free space do you have on your system drive?

When is the last time your repaired your disk using Disk Utility?

Allan

Also, how much RAM installed?

Also, check to see how many items and starting at Login.

You can find Login Items under the System Preferences pane under the Accounts icon.

Have too many login items can make your iMac take longer to boot up.

See:

Mac Maintenance Quick Assist,
Mac OS X speed FAQ,
Speeding up Macs,
Macintosh OS X Routine Maintenance,
Essential Mac Maintenance: Get set up,
Essential Mac Maintenance: Rev up your routines,
Maintaining OS X, 
Five Mac maintenance myths, and
Myths of required versus not required maintenance for Mac OS X for information.