(Semi) newbie here with many, MANY questions!

[Snagrio]

I have one of these that have decent reviews that are within your budget and they come with a probe. They're also made by Vivarium Electronics that make proportional thermostats that have been a good industry standard. I'm sure you can do some googling regarding people's opinions:
https://www.reptilebasics.com/thermometers/TH-100/

Ah. I actually saw that one in particular on a YT channel and thought it'd be a good fit, but the person who made the video messed up the link to it in the description so I couldn't find it. Thank you!