An all-in-one will present less of a learning curve, but also provide less flexibility in choosing quality components.
https://www.amazon.com/Zoom-Q4-Handy-Video-Recorder/dp/B00GWDCA8E/ref=sr_1_1?ie=UTF8&qid=1451331692&sr=8-1&keywords=zoom+video+recorderhttps://www.amazon.com/TASCAM-DR-V1HD-Video-Linear-Recorder/dp/B008PAI770/ref=sr_1_8?ie=UTF8&qid=1451331692&sr=8-8&keywords=zoom+video+recorderIf you separate the video and audio, you will be able to independently position each in their optimum place. Having the mic on camera is often not the best thing, as it can pick up operational and handling noise. Look at some of the budget condenser mics that can be had in the $100 range (you will need two to do stereo). Mic positioning and stereo technique will be a large component in the outcome of your recording, so don't focus on gear alone.
With the condenser mics, you will need a preamp with phantom power. You can get audio interfaces that hook up to your computer with built in mic preamps. Naturally, you will need one with at least two channels to hook up with your two mics. Some budget level ones start at $100. Keep in mind that some of these interfaces also run in the same price range as some portable recorders with built in XLR inputs and preamps. So there are decisions to make.
Recording piano with SM58s is very tempting due to their availability. However, piano is not their application strength. Please consider other options.
You could get one of those portable recorders with built in mic preamps and have more options for mic placement with the addition of external mics. Generally speaking though, you will get better sound from stand-alone mics as opposed to those integrated into a recorder.
For video, a compact camera with 1080p 24fps would do fine. No need to limit yourself to purpose built "video cameras" as still digital cameras do video quite well. DSLRs are also great for video, and some have inputs for external mics. Be careful when reading camera specs. Some early 1080p cameras did 1080p at 20 fps (or similar low frame rate), just to advertise that they could. You need a minimum of 24 fps, or you will get choppy video. Nothing wrong with 720p, just realize it is a lower resolution. Go search for a used camera gear, good deals to be had.
An older model DSLR with kit lens that does 1080p/24fps or 720p/30fps for around $200-300 is great. Larger sensor then compact cameras, choice in lenses, scalable system with large range of accessories.
Modern cell phone video isn't bad. However, people wielding these devices often use poor video technique and thus make the video much worse then it could be. If you are using a cell phone as a video camera, treat it like you would a real video camera. Do not shoot in portrait orientation, just don't. Video needs to be shot landscape. Use a stabilizer or a tripod.
Tripod: get a tripod, you need one regardless of what camera you use. Don't even think about doing video without a tripod. A table and a phone book is not a tripod. Get a tripod. The key check on your piano is not a tripod. Get a tripod. Your friend's hand is not a tripod, unless they have a stabilization rig. Get a tripod.
Software:
Agree with the Cool Edit recommendation above. Adobe Audition v1 came from Cool Edit and behaved much like its predecessor. Later versions of Audition were progressively Adobefyed.
https://audacityteam.orgAudacity. For wav editing, start here first. For many people it does everything they need. Can't beat the price either.
https://ardour.orgArdour is an open source DAW. If you want something similar to Cubase, Sonar, Pro Tools, etc. without the price look here. Linux and Mac only.
https://www.reaper.fmReaper is an affordable commercial DAW. For a small price you get a DAW with the essential features you need.
As for NLEs I am still searching for a good free or affordable solution.