Modern phones actually aren't all that bad at making a somewhat decent recording.
It's about the placement of it.
If a recording sounds bad from your phone, it is possible that you are too close to the mic when recording, making it distort and clip the louder frequencies of your sound. If too far, the recording might be too vague and faint. You'll have to experiment with distances and location of your playing relative to your phone to get the best results.
No need for apps. Just use your phone's video recorder or sound recorder.
You can easily upload and share recordings on websites like Youtube and Soundcloud.
I find Youtube the easiest platform to share on. If you don't want to show yourself on video, either record with your camera covered or it pointed away from you. Or just edit the video so there is some image or title screen showing while the audio plays.
That being said, for better results, you should consider getting a microphone and recording on your computer.
A USB Condenser microphone, like a Samson C01U, Blue Yeti, Blue Snowball, etc, do not cost very much but will be a huge upgrade from a cell phone mic.
You can either record on your computer's default sound recorder or webcam recorder OR get an audio recording software (Audacity is free and very good, though it takes a bit of getting used to), and record through that.
For ideal results, you'd have to invest in more sophisticated equipment like a USB interface for XLR mics, and some studio grade microphones, etc. For an even more ideal result, you'd record in an actual recording studio. But you don't have to go that far for some decent home made results as per the methods described above.
EG: I recorded this with a Samson C01U USB microphone which retails for under $99 on eBay, using Audacity sound recording app:
https://www.youtube.com/watch?v=cNyuI3evT44
I recorded this on on my computer's default webcam software using the same mic:
https://www.youtube.com/watch?v=gXoCvUWtwzc
For comparison, this is recorded purely on my Samsung cell phone:
https://www.youtube.com/watch?v=USX_cL0dDXc