Becoming a sofer
To be a sofer, you have to master two areas of study: halakha and practical.
If you're not interested in the halakhic part, you can still make pretty calligraphy. That's okay. The world needs calligraphers. You just won't be a sofer, because there is no safrut without halakha.
You have to study. You can't just sit in a class for three months, write a paper, and graduate. You've got to own the material, all of it. That means that a huge part of learning to be a sofer is entirely up to you. You have to work on your calligraphy and practical skills and you have to learn the halakha. A teacher can show you techniques, explain things, help you over the hard bits - but most of it is up to you.
Especially, you can get yourself started.
If you are totally new to calligraphy, you will need to learn the basics. There is a lot of information on the basics out there, but it's mostly in an English context. This is fine; the basic skills are completely transferrable. In fact, most of the Hebrew calligraphy books out there aren't ideal for beginners anyway. Start with a marker (try these ones or these ones) and printer paper, and refer to calligraphy websites; progress to more expensive tools and books as you develop.
"But I have terrible handwriting!" - With practice, you can become quite a reasonable calligrapher. It all depends on how hard you want to work at it.
Lefties: you have it harder learning English calligraphy, but you will have a fantastic time once you start doing Hebrew.
Learn to walk before you can run. Develop the motor skills which will enable you to form neat, consistent letters. Learn to use the different kinds of writing tools; felt-tip pens, fountain pens (skip this step if you are unnaturally precocious), and dip pens. Learn an easy English alphabet (such as Uncial) or a simple Hebrew one. Gain reasonable agility doing this with dip pens before advancing to quills and the sofer's aleph-bet (see below). This stage is time-consuming and not all that exciting. C'est la vie.
A sofer works on parchment with kosher ink. You can use these materials to learn on, but they are immensely expensive, so you are better off starting out with ordinary calligraphy ink on ordinary printer paper.
When you are ready, move to learning the letters for safrut. Learn the letters and the halakhot of how to make them simultaneously, otherwise you will form bad habits. Here is a worksheet, and here you can find videos. From this point, having an expert look over your work now and again will help you, but it is still mostly about practice.
Learning to cut a quill: here is one video (free), and here is another (not free). Here are several websites. You can buy materials here. Now practice a lot. Use Google to learn some basic knife techniques, they will make your life easier. Be careful with knives.
Once you can make decent letters and cut a reasonable quill, you are a calligrapher. You cannot be a sofer until you have also mastered the halakha.
For basic certification, you need to know the rules in the Keset ha-Sofer (buy online/download 3Mb PDF/download 36Mb PDF/English) and the rulings in the commentary the Lishkat ha-Sofer. You don't need to memorise the details of the back-and-forth in the Lishkat. You also need to know the Mishnah Berurah on simanim 32 (Hebrew, partial/English, partial) and 36 and Mishnat Soferim, and the rulings of the Beur Halakha as for the Lishkat ha-Sofer. Here are videos of the basics, in Hebrew. Here is an interesting question-and-answer board.
You need to be able to handle all the above in the original Hebrew. If you are totally new to rabbinic texts, this is going to be a challenge, but you can do it if you work at it. A couple of years in yeshiva would be the ideal way to set about it. Otherwise, you may find a sofer who is willing to take you through the texts, or a learning partner who will help you get a feel for rabbinic texts (in person or virtually), or you can tackle it alone. Much of the material is translated into English online; this helps many people.
If you have the tools to learn halakha, learn as much as you can, and find a sofer to help you with the rest (google "learn safrus"). Some of it isn't tremendously obvious.
Calligraphy + Halakha = Safrut
Now you need to find a real live teacher to help you put the skills together and create sifrei kodesh. If you are a man, this will be relatively easy. Use Google, contacts, and common sense. If you are a woman, it is less easy, but stick with it. Emailing me is a good place to start.
How long it takes depends on you. If you are a good calligrapher and fluent in rabbinic Hebrew, you can become minimally competent in weeks or months. If you are starting from the beginning in both, you can become competent in a year if you put enough time and effort into it. It will take longer if you put less time into it.
Note on personal development
Part of the halakha deals with who may be a sofer. In brief, a sofer is entrusted with transmitting the Torah, which represents the community's highest ideals, and a sofer must be worthy of that trust. Whatever community you choose to work in, you must have its respect. If your Jewish practice isn't all it might be by your community's standards, you need to start working on that. They must be able to trust you to transmit their Torah and its ideals safely. You have to demonstrate yourself worthy of that trust and in tune with those ideals.
I believe that in order to write kosher sifrei Torah, tefillin or mezuzot (but not the Megillah, which is different) you must also have a strong commitment to the mitzvah of tefillin, no matter what your community's position is on tefillin. Our most ancient rabbinic tradition links being permitted to write sifrei Torah very closely to the mitzvah of tefillin. They are deeply, if mysteriously, connected. If tefillin are not part of your Jewish practice, I strongly suggest that they become part of it forthwith.