These questions are probably too hard to answer as they are too general without knowing more about you. As a career you can make a good one out of both Graphics/Multimedia or Programming. Are you after which makes the most money or which will give you long term satisfaction in your work? These things can change from country to country too, so what is true for me here in NZ might be terrible advice for someone based in Australia even though the countries are close and quite similar.
There are huge amounts of variation in what kind of day-to-day work could be done in any area too. I have a background in using multiple types of 2D design software. I've been doing it for a long time successfully earning a living from it, but my immersion into 3D for a recent job showed me how specialised that work is, and how it doesn't come naturally for me to think spatially like would be required. There are aspects of it that overlap and would be easy for me, but I could see there was also another part that would be very difficult. This is all to say that you need some good advice and to spend some time exploring these careers to make sure you get a good match between your skills, interest and capabilities.
There are lots of trial software or free & open source software that will give you an insight into some of these areas without having to invest in training until you are sure you know what you are getting yourself in for. As always a qualification will open some doors more easily but isn't the only way to make a living. There are some very skilled people who have never been formally trained through an institution.
Which is the best career option Graphics & Multimedia (Animation) or Programming of Software ?
Also, which software is required for creating a game with graphics and multimedia options?
Career wise, you'll need to explore more options but a lot of people just stay in freelance work if they have the graphics skill set.
As for software, the basic set of software you'll need is everything Adobe offers:
Photoshop, Dreamweaver, and Illustrator are the most common ones you'll see yourself using.
Graphics and Multimedia stream offers endless career option to the aspirants.There are numerous career options such as VFX artist,Graphic Designer,3D Animation artist,Web designer etc.According to me Web Designing would be a good career option for those who have passion for programming and designing.