A great place to start is the mini painting starter sets by Reaper:

https://www.amazon.com/Reaper-Miniatures-08906-Learn-Paint/dp/B00NTMC49G

https://www.amazon.com/dp/B01N458GBK/

They're not too expensive, and they are a great intro to mini painting. They also come with pieces to practice techniques on, and a lot of paints to work with as well. I'd get those, and maybe some brushes.

I also would recommend a wet palette to keep paints wet between painting sessions, magnifying glasses to help see, and a mini holder.

I also got these brushes, which seem to be working well for me, and have a mix of useful ones.

I got the higher-tier version of the set you linked (the $150 box) and have been fairly happy with it. You don't need too many paints if you're just starting out, and you can always mix paints together to get shades you don't have (or close to them).

Before spending your entire budget on paint, I'd add some important stuff to your budget - a high quality brush makes all the difference (A Kolinsky sable Round #0 or #1 should be good enough for everything, https://www.amazon.com/Winsor-Newton-Kolinsky-Sable-Watercolor/dp/B0013E68SU?th=1) and stainless steel or hematite mixing balls for your paints (I use https://www.amazon.com/gp/product/B07Q23FVYR).

The Reaper learn to paint kits (Basic and Layer) have pretty dang good paints, as well as 6 minis to learn techniques on. The variety of paints they come with should just about cover what you need to get a decent paint job on Gloomhaven stuff.

Last but not least, I would pick up some Nuln Oil shade. It's pretty much a staple in every mini painter's collection. I'd look for it in a local hobby shop if they sell Citadel stuff; I can get mine locally for $7.80/pot, versus the $13+ on amazon.

All told, that comes right around $100 for paints, practice minis, and some very good supplies.