Hey! I made a little rundown in the past on the various options available for video software. Here it is:
Have Apple and money? Final Cut Pro.
Have Apple but no money? iMovie.
Have Windows and money? Adobe Premiere Pro (What I use 90% of the time).
Have Windows but no money? Lightworks (complex) or DaVinci Resolve (simpler). Even Windows Movie Maker can accomplish basic assembly, although I don’t believe you can overlay extra audio tracks. So you’d have to be assembling a combined A/V file (Cell phone video, essentially).
You can also use GarageBand or Logic Pro on Apple for audio-only and on Windows you can use Audacity, Pro Tools, or Reaper (the bigger 3).
As far as hardware, I’ve recorded everything so far on my cell phone, and replaced the audio with what I recorded with my microphone setup. If you don’t have that available, you can also set your cell phone leaning against your amp and record that way. If you set it against the amp, you can get a very nice cleaner sound because the relative loudness of the amp will overpower extra noise such as your picking, etc.