Some good suggestions here already, and I'd like to add the following...
The first things to ask yourself are:
1. Do you want to record to a standalone device, like one of the Zoom, Tascam or other digital recorders that have a mic or two built-in, and use an SD-card?
or
2. Do you want to record to your phone/tablet with a one or two small add-on devices?
or
3. Do you want to record directly into the computer?
and
4. Using any or all of the above do you plan to make videos in the future, for YouTube or the UU Seasons weekly themes?
All of the above can be combined in various ways, and yes with free software (such as Audacity), but if simplicity is your goal and you prefer not to deal with a mess of wires, I can recommend the following starting points based upon having a home recording setup since 1985 and working in pro studios on and off since 1990:
1. For $99 US, you can get a Zoom H1 that sounds GREAT with the built-in mics (I just got one), and can take other dynamic and electret mics, as well as the input from your guitar cable (and the pickup in your uke if it has one), records to a microSD card, and can ALSO function as a USB mic into the computer and/or directly into an iPad or iPhone with the USB camera connection kit and a powered USB hub.
I have not yet tested it on Android, but USB class-compliant audio on Android is a bit of a fustercluck, despite this being part of the Linux kernel for over 10 yrs now, so I would avoid Android unless you want to spend $40 for the proprietary USB audio drivers that only work with 2-3 apps.
2. For $199 US, you can get a factory-refurb Apogee MiC direct from Apogee. The new model does 96k, which unless you are listening in $500 headphones, or with professional studio monitors, you will NOT hear the difference from the original model that is only 48k.
I also have the Apogee MiC and this is pro-level hardware and sounds GREAT, but is MONO and does not have a headphone output. It comes with cables for USB and iOS devices, and works with all the above without any extra parts.
I have personally tested it on Mac and Linux and and iPod Touch and 2 different iPads and can testify that it works well.
I bought the refurb model myself 3 yrs ago and have never ever had a problem with it.
Someone is likely going to recommend the Blue Yeti microphone which is a very good mic, but the Yeti is physically HUGE in comparison (about 1ft high and 7" diameter desk footprint) to both the Zoom H1 and the Apogee MiC. The Apogee MiC is about the size of a standard 'D' sized battery. Using the Yeti with iOS devices requires a powered USB hub as well.
Links are below:
Zoom H1
https://zoom-na.com/products/field-video-recording/field-recording/zoom-h1-handy-recorder
B and H Photo in NYC has them the cheapest right now in different colors, such as a limited edition blue model for $69:
https://www.bhphotovideo.com/c/sear...ts=ps&Ns=p_PRICE_2|0&N=4291104436&Ntt=zoom h1
Apogee MiC:
if you scroll down you will see the original Apogee MiC refurb for $169 and the updated 96K model for $199:
http://www.apogeedigital.com/store
I have discussed the Apogee MiC on UU many many times and if you search for 'apogee' (sans quotes) you can find those discussions.
As an audio engineer being my second job (IT consultant is primary) for so many yrs, and with the performance of these devices in my own hands, I highly recommend that you skip over the cheaper options.
However, the cheaper options will 'work', to capture audio, and of course the sky is limit if you want to spend more...
As far as recording methods, etc using anything mentioned here, that is a bigger conversation, and I have a few YouTube channels that are quite helpful if/when you are ready...
As far as
video, I am planning to evaluate the Zoom Q2n soon, and am hopeful that I can recommend it, but without having hands-on myself, I am not giving it my blessing just yet, but if you are curious, for $169, it solves a lot of problems for a musician who wishes to make videos.
See here:
https://www.zoom-na.com/products/field-video-recording/video-recording/zoom-q2n-handy-video-recorder
Any further questions, just ask, and I am happy to answer.