- MUSIC -
Many people use
FL Studio, myself included (though I also use Sonar), and it's perfectly capable of making professional quality music of any genre for any media purposes (well, perhaps not surround sound...).
Since you don't seem to have a midi controller, FL would probably be a great choice, as it allows you to use your computer keyboard to play midi notes. It's not nearly as nice as a midi controller, but I used my computer keyboard before I had a midi controller and it's not bad at all. Of course, you can also manually draw in the midi notes like any other DAW (digital audio workstation).
But FL Studio isn't free, so I'm not sure if that's a deal breaker for you. That said, there is a free version and it has almost no downsides - no expiration and next to no limits on plugins AFAIK, but the downside is that it won't allow you to open project files; you can save your project, but you can't open it. This means you can do a single song in one go in the free version, leaving FL open as long as needed to complete your song, and you can save the project for future purposes, if you ever end up buying FL.
It can work great if your computer is always on and you're the only person who uses it, but it's probably not ideal for most people.
FL also has a sound font player, and if you search around online, you can find pretty high quality sound fonts for virtually any instrument and many of them are totally free to use for any purpose.
An alternative to FL might be
LMMS, but I wasn't impressed by it, personally, though it's totally free, so that's cool. But I don't remember if it allows using a keyboard to play midi. If not, there's always the manual piano roll where you draw in notes with the mouse.
Both of these DAWs can work just fine without requiring an audio interface, which may not be the case for other DAWs. I recommend these since it sounds like you don't have any music production hardware.
- SOUND EFFECTS -
I haven't used FL Studio to make all that many sound effects just yet, but I still feel pretty confident in saying that it's also good for making sounds, since it comes with some plugins which can be good for sound design, but it can get a bit technical, needing at least a basic knowledge of audio science and sound design. You might just want to go for
bfxr or
ChipTone.
If you're looking for less 8-bit sounds, I'm afraid I haven't done enough research to recommend anything for good original sounds without using a mic and a good plugin that's capable of resynthesizing, like Harmor.