I have a small IT shop where I do exactly this with Redmine. Tracking my work and collecting information about my tasks.
Redmine isn't just tied to software development. It's just the terminology like "target release" that makes it look like it is.
To start, I would do the following:
- Collect your task categories and create trackers for them.
- Design the workflow for each tracker. Create custom states as needed. (Keep it simple. New -> In progress -> Done is sufficient for most tasks).
- Specify form fields as needed.
And I highly recommend playing around with Redmine at least a little.
There are virtual machine images from Bitnami or Docker images that you can use to spin up an installation and start playing around with Redmine.