Being able to write a song isn't something that most people can just 'do' straight off, so don't be discouraged if you're not happy with your early attempts. It's like painting or any other type of creative outlet- you need practice to get better. It's okay if your stuff sounds cheesy, as long as you write from your heart and personal experiences or thoughts, and don't just try to make it sound like other songs. Don't feel pressured to stay within the common 'verse-chorus-verse-chorus-bridge-chorus' structure either- a lot of really great songs don't follow that pattern, though most pop songs do.
The most important thing in songwriting though, is learn an instrument. You don't have to be particularly skilled, but it's extremely hard to write songs without being able to sound them out as you go. Guitar or piano are both great choices, and you'll be able to start writing after you've learned just a few chord progressions. You can pick up incredibly cheap second-hand guitars too, and there are so many resources online.
As for recording, a great starter mic is the Blue Yeti. It's USB, and has a nice frequency response to both vocals and acoustic instruments. The price is incredibly awesome for the sound quality and ease of use. It's compatible with any audio software I can think of too. =)
I hope this is helpful- let me know if any of this raises questions. I'm always happy to help a fellow budding musician. :3 Good luck!
(feel free to pm me if you want a listen of the Blue Yeti in action - I have some old recordings somewhere)
Come chill with sound sometime, you're always very welcome. 
