I would recommend to use the official Oracle® Database Upgrade Guide 10g Release 2 (10.2).
Be sure you have enough space and install the 10.2 Oracle software on the machine in a new/separate Oracle Home. I typically use /u01/dba/oracle/product/10.2.0. Once the software is installed follow the suggested migration guide to upgrade individual database instances. I have numerous servers which have multiple versions of Oracle running on them with no conflict. Whether you use 1 or multiple listeners will need to be decided, but if you aren’t using Oracle Names, then using the highest version for the listener is typically the best idea.
Be careful with your environment variables and follow the guide.