Should I become a programmer?

The fact that you're reading this most likely means you should.
Programming is one of the better paid jobs depending how good you are how much experience you have and it also has the benefit of not being as stressful as a graphics designer for example, however depending on your mental personality being a graphics designer might be better for you.
First off I'll start by saying that according to science, you should first consider the areas you are better at.

Studies show that while some people process math better, others process images better.
First off you should think about what were you better at in school?
Was it math, or literature and painting?
Are you better at puzzles, or are you better at writing, reading, etc.
Being good at math by no means makes you a good programmer. You can be very bad at math because of abstract numbers which offer no satisfaction in getting a result. Some people want something more palpable.
So another way to look at it is. Are you a logical or emotional thinker?
If you are a patient, logical thinker then programming might definitively suit you. Yet if you are more into the visual/emotional side, then you probably will get bored of programming and quit it or not be very productive at it.
This does by no means imply you can't program. The problem lies in doing it constantly for a living, you must be prepared for that.

If you still want to go with programming then your next question will probably be where do I start?

Well you could start by learning any programming language really. Even a web programming language like php.
Probably python nowadays seems to be the more often recomended choice.
Basically look around what calls to you more and what seems to make more sense to you and google a tutorial for it.
Remember web development is programming too, now a lot of stuff is moved to the web development area.
So do not think "Ahh anyone can do websites!" Well yeah. Throw in some easy html some css generators. But when it comes to php, javascript and making custom client-server interaction for example things get tricky, and this job will pay just as much.
In fact I would encourage trying web development first, it's easy, it has a ton of tutorials, and it's a great start for anyone looking to get into this field.
Once you grasp the concepts of programming from either php or javascript, you can move on to more complex languages like c++ or rather since you might want something new, even though c++ is still widely used on everything, learn java, or c#. or if they seem to complex to you when you get to make actual programs, start off with visual basic.
And learn, try, don't be afraid to screw up, eventually it will all start fitting together, and it will take more than one week it's not something you learn in a day. No.
Don't stick to only one programming language either, explore, whatever is easier for you is probably your best option.
There a lots of tutorials however I suggest after getting past the basics, try using those basics to make a local database, add users, passwords, add functionality, interaction. Then you will start understanding how to piece everything together. After that you can start to actually make simple yet useful programs.

All photos are public domain