Mic and Mixer Reccommendation?

Hello!

Hobbyist musician here. I sing and play an acoustic electro guitar, and electric guitar occasionally. A fan of 00s rock/indie/folk songs. And there's a lot of songs I want to perform with live looping for both vocals and guitar parts. And I was wondering if anyone had any suggestions for a:

  1. Microphone that's good for vocals (performing and recording) but can be plugged into the mixer without a pre-amp.
  2. A mixer to plug the mic and the guitar into, as I was thinking for a set up like this?

https://preview.redd.it/0so49zhaqfpe1.png?width=973&format=png&auto=webp&s=ce09d1030dda65ca1c2e2477daa8fb44a33dbd40

Feel free to let me know if that set up even is any good or any suggestions that could be better! Super new to all of this, I mostly sing songs at home but I've recently signed up to a casual open mic which will be outdoors so I'm not shooting for the stars quite yet, I know it's gonna be pretty amateur.

Fairly lower budget, not looking for any gear above £200 individually and doesn't have to be super fancy with lots of features.

Thanks! Any advice is appreciated.