If you have never done vector graphics, Inkscape is a great place to start. Inkscape is Open Source software, meaning you can download it and try it out. The creators ask for a donation but this means laying out far less money than for proprietary software.
The wonderful thing about Inkscape is that not only is it a good starting point, primarily because of price, it is also a program that can take you into doing professional-quality work. It has most of the features of other (high-priced) software. If you look around on Youtube and Google, you can find professional designers who use Inkscape and who can tell you how to make Inkscape do what its expensive counterparts do.
Some people fear downloading and using Open Source software such as Inkscape because they worry about technical support. Inkscape has a wiki, a webpage created by many Inkscape programmers and users, with a lot of information about how to use Inkscape and how to solve problems with it. The learning portion of the Inkscape website has video tutorials and other information. There is also a community forum which offers several ways of getting in contact with more expert users.
Additionally, Inkscape has been around long enough that there are many resources across the web for learning what you can do and how to do it. For example, there are videos on Youtube from a range of people.
Inkscape can take you from the trial phase of learning about vector graphic design all the way to being able to do that professionally, and all this for a great price. Review collected by and hosted on G2.com.