I would start by looking at these two videos, they told me a lot. In their examples the artist plays a guitar, but I assume it translates to ukuleles.
Also, they naturally suggest which Neumann mics to use, since they are trying to sell those. But I guess the placement also works well with other brands, or with a dynamic mic in stead of condensers.
I have only ever used the one mic for simultaneous vocal and uke method. It has been for video, where two mics placed in proper position would be bad for the shot - especially if I use a reflection filter. If you use that method, you have the option of using a USB mic or am XLR microphone - which require an interface, which is the hardware you plug the microphone into. You cannot just use and XLR to minijack adapter, because you also need the preamps etc. from the interface to get any sound out of it.
If you use two microphones, you will need the XLR mics.
If you are on a budget, and dont have a newer laptop to process the music, you could just stick to the Yeti for a while. Perhaps invest in a mic stand, a better DAW, like perhaps Reaper, or some other stuff.
Within USB mics, I have used a Yeti and a Shure MV5. Of those two, the Yeti sound better to me. You can get better USB mics. The AT2020 and the Røde NT USB will both be a bit more serious according to reviews and sound samples, though they dont have stereo like the Yeti. They would set you back about $200, unless you find a second hand deal. A lot of people bought stuff like this during the covid lockdown, and are selling. Notice that the Apogee mic mentioned in another post might only work with Mac/Apple products.
Within XLR mics, I have tried a SE2200 large condenser mic and a Røde NT3 medium size condenser mic. They are pretty similar. I have a slight preference for the Se2200. I use them with an Audient iD14 interface. Less can do it, there is an abundance of options. If you want to be taken serious be musicians, you need XLR and interface. Not only because of sound quality, but because even the best USB mics send the message that you are not really a musician. For that reason i stuck with USB for years, to keep peoples expectations low
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Regardless of what you go for, you will need to calibrate the gain to avoid clipping. But once you have found the settings that work with your setup, you can pretty much leave them there. As long as you record with a bit of headroom and add gain afterwards, small loudness changes in your singing should not differ too much from song to song.