The only trouble you might have recording directly into the PC with either the instruments or microphones would be with low recording levels and excess noise due to the nature of the internal PC sound card. Also, you will want to use the line in (if it exists) for instruments rather than the mic input. If you are just recording the songs for your own purposes or to share with family and friends, this method will work just fine. If you were looking for a more professional sounding recording, you will probably need to invest in some additional hardware.
To get the best sounding recording you will want a USB recording interface. Just about any of them will give you a cleaner recording than the internal sound card. You will want one with at least 2 inputs if you want to record the keyboard in stereo. You will also want to make sure that it has MIDI, as some of the less expensive ones only do audio. I would recommend trying one from M-Audio, as a good starting point. They sound great and are about mid range as far as the price.
As for the software, Reaper will probably be your best bet. Reaper is a full featured DAW which will allow you to record MIDI and audio. You can add effects and add as many tracks as you need.
I am currently using an M-Audio interface with Reaper to record Keyboard, Piano, Drums and Vocals. It can work for you too :)