Comment My few cents... (Score 3, Informative) 144
There are already a lot of suggestions for which particular package to use, so I'll contribute my thoughts.
I've used RT. It worked.
The main feature that helped me move my (financial services) office from word-of-mouth to proper tracking is that RT supported submitting issues by email. We already used internal email extensively for other workflows, so it was easy for me to convince people to send an email to <project>@tracking.<domain>, and they'd get an emailed response showing the ticket number and a link where they could follow the progress.
RT let me run different projects (which in my case usually meant only 1-3 people) separately, and each project had nice charts showing how many open issues they had to work on. Managers loved that, because they could see who was overloaded and by how much. Each user was also able to create their own dashboard to display when they logged in, so they could get a to-do list first thing in the morning.
For each project, I could modify what information was needed when a ticket was created. Almost the entire ticket form was customizable, so that was adapted to the project needs. For our financial advisors, they had simple forms with a customer name and a description field. Traders had buy and sell forms with security symbols, amounts, account numbers, et cetera.
I haven't been in a position to manage very many trackers, so maybe these features are standard-issue. Maybe something else will work for you, but like it said, RT worked for my needs.