The terminal release for Oracle8 was 8.0.6, ensure you upgrade to the terminal patch release 8.0.6.3. The next step would be to Oracle 8i, the terminal release for that was 8.1.7.4
This info comes from OTN http://www.oracle.com/technology/products/index.html - select ‘Database Standard/Enterprise Editions’ from the section ‘Database’. That will lead you to Metalink links for which you will need a login id.

[[Can I upgrade a test database first for testing purposes, and once satisfied upgrade the production database ?]]

This is a good idea to give you a feel for the work and time involved. All the better if your test database conforms to your production database, then you can test functionality. Going from 8.0.5 to 8.0.6 is not a big step. I would advise reading the release notes and info about the new release to find out about new and deprecated features.

[[Is this just done by setting the correct Oracle_SID and running the upgrade process?]]

This is the question which would take all the time to answer - read the documentation. 8.0.5 > 8.0.6 is not a big step (more or less a patch in fact) but there will be documentation to support it with the software.

[[Do I have to run multiple listeners (8.0.5 and 8.0.6 )?]]

In my experience, no. A later version of a listener will work for earlier database versions.

[[To upgrade each instance do I have to run the upgrade procedure for each instance?]]

Generally yes. Upgrades often include Data Dictionary changes. If you have to run code from SQL*Plus or Enterprise Manager, this indicates that the code does something to the instance, so will be required for each instance.

Sorry for the format of this reply, but there is not a lot I can do about it!!

HTH