There are plenty of tools out there that can help with time tracking. Start by doing a Google search on "time tracking software." I suggest you choose one that allows your team to track and enter their own times by categories. Break down the major tasks that a DBA does, and create separate entries for each. Be careful not to get too specific though, as often this only creates confusion. Some example high-level categories are monitoring, troubleshooting, change management, installs, upgrades, maintenance, etc. Keep in mind that you also need to create non-DBA categories for the time spent doing non-DBA related work. For example, training, meetings, research, documentation, etc.
As for change management, if you have a ticket or help desk request tracking system, start with that. The most important thing is that you define and document your processes and get your team, as well as the users they support, in the habit of creating a change or service request for every change. If there is going to be an approval process involved, you need to make sure the software you choose allows for that. We base our change management on the ITIL (IT Infrastructure Library) framework and use a third-party online case management system. Using SaaS (Software as a Service) is a great alternative to buying or building your own solution.
There is a plethora of information on the Web about ITIL, but your best bet is to buy the official Service Support book (blue book) and cover the following chapters in detail:
4. The Service Desk
5. Incident Management
8. Change Management
9. Release Management
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