I'm an amateur singer/songwriter who only recently started writing music. Before that, I only knew how to sing (choir) and play some basic guitar chords. I really didn't know anything about scales (still don't have a solid grasp on them), I didn't read sheet music (I still don't!)-- I only really had that desire to create music. In short, I'd say I was at a stage similar to where you are now.
Fast forward about a year - and I've been able to write and record (demos) a fair number of originals. Still a long way from professional, but both the results and process of my songwriting make me happy.
Here are the things that helped me in my own journey:
1. Keeping a scratchpad of thoughts and tunes.
I keep a journal for writing memories, rants, dreams, inspirations, random thoughts, etc. I did this even before I started songwriting-- eventually I realized that it made a good repository for material for writing lyrics. In addition, whenever I get sudden bursts of inspiration for a tune (this usually happens in the car!), I make sure to record it right away. I just use the voice recording utility on my phone or laptop, whichever's at hand. This way I have a repository of little song snippets.
2. Having a convenient songwriting tool.
In my case this is Garageband on my laptop! The interface is intuitive enough for me to be able to easily compose songs even without knowing how to read sheet music. I don't mean to say that this is the "best" music composition software or anything-- it's just that I personally found this to be the most convenient tool to use for my own songwriting. Each composer will have his/her own favourite tool, and it doesn't even have to be digital, it could just be a guitar and a pencil and paper. It just has to be something easy and comfortable for you.
3. Learning and performing my favourite songs.
"Good artists copy, great artists steal." Every time you study a song and practice performing it, you can learn something about rhythm, melody, and/or harmony, plus you develop your "ear". I don't know if I'm wording this correctly-- but when you develop your "ear", you're able to notice more subtle nuances in the music, and you can tell if it sounds "right" or if it sounds even slightly "off", and then you can figure out why and fix it.
4. Taking voice lessons.
I'm assuming that when you write your songs, you intend to have vocals? If so, I would also advise taking some voice lessons (or at least regularly vocalising with scales), even if you don't plan to perform the songs yourself. Voice training develops your "ear" as well.
And lastly, this is cliché but, I think the most important thing was that I just kept at it, and I put in the hours. I agree with the answers above, start simple, and writing something bad is better than writing nothing. Good luck and enjoy! :)
P.S. Glad you asked this question because I'm learning a lot from the answers here too.