Concentrate on writing one genre or many?
So you want to write video game music for a living but you don't know if you should concentrate on learning to be a specialist or a generalist. Now let me get this out of the way, if you are writing for TV or Film I would definitely recommend sticking to one or two genres and becoming an expert in those as this will give you the reputation as the go to person for them. As a video game composer though, it maybe a better idea to generalise first before specialising. Let me explain why.
Early Work You'll Get As A Video Game Composer
When you start out writing music for games it is likely you will be writing for individuals who are trying to make a small game with the occasional small team hiring you. At this point in your career you will need to be very adaptable to your clients wants. You will probably have to accept any work you can get and, in my experience, there will be such a variation in work you cannot predict what genres to get good at beforehand. With video game music, you have to learn on the job, so what can you do to prepare?
How I Would start Out
First off, I would be listening to video game sound tracks all day everyday. This will prime your mind for what you may get asked to make. Then I would choose a selection of tracks from each genre to either recreate, or use as inspiration to write your own music. You won't be able to prepare for every scenario (I had to make a soundtrack with Kazoos once for instance) but you should be able to competantly create the reference music in a few hours before moving on.
Next I would join some freelance sites, Fiverr being the most obvious, and set up some gigs advertising my services. At the beginning, set your prices low so you attract the most clients and see what comes in. Try to accept anything and everything unless you know 100% you cannot do the work. The main thing to do at this point is do the work to the best of your ability and learn on the job. Remember, if you really mess up at the beginning, it isn't the end of the world. You can always start again if you have negative reviews, but you might suprise yourself as I did with 5 star after 5 star review, all which will gain you testimonies you can use in the future to get higher paid work.
What if I really Want To Just Compose 'X' Genre?
If you are really dead set on say, making just cinematic video game music, then this is how I would approach that. I would obviously be writing music for this genre using reference tracks every day and I would also create freelance gigs, but most of all I would be creating social media channels to specifically show off my work. In this instance I may also decide to try and get employed as a composer for a video game company as you can select one that makes game with the specific genre you make in mind. Working on showreels and having a good website to direct clients to will also be advisable whether you get hired or for a company or end up going freelance.
I hope this gives you a general overview of how I'd approach this topic. Both generalising or specialising has it's pros and cons, but I decided to generalise so I could accept the most work possible and, unless you really want to make a specific genre, I'd say this is the best option for most composers in their early career.