First, if you look in the upper right-hand corner of this web page, you will see a white box labelled "search". If you click on that box, type "automation", and hit Return, you will see links to many questions and answers about automation. I believe it will be worth your time to scrutinize what you see there.
This is a big subject, but if it were me, I would first decide whether my primary motivation is to save time or find more bugs.
If your primary motivation is to save time, you should start by automating some common, time-intensive setup tasks, for example, generating test data or automatically installing and configuring your product. These things are a good place to start because they are measurable, which may be important to your management. If you can report quantifiable success with this, it may be easier to justify additional kinds of automation. In my experience, automating setup tasks also requires less programming skill than automating tests.
If you have serious quality problems now, and you suspect that automated tests may help you find more bugs, you should identify some specific areas where automation may help you. There are many strategies for identifying tests to automate, e.g. areas that are particularly buggy, areas that are particularly time-consuming or error-prone to test, or areas with a combinatorial set of possible inputs but which have easily-predicted outputs.
Be forewarned that despite the claims of tool vendors and automated testing evangelists, automated testing is not a magic bullet. Moreover, like all software, automated tests require an up-front investment followed by ongoing maintenance. Approach automation as you would any large project: start small, be honest about your results, and be prepared to switch tactics if you are not achieving your goals.