I'd recommend this set from Reaper and they have another called Layers I believe. https://www.amazon.com/dp/B00NTMC49G/ref=cm_sw_r_cp_apa_i_1YXgFbZF3B5BK
Those will get you making some damn good mints and teach you some good skills.
Then pick up some shade. https://www.amazon.com/dp/B0189BIERC/ref=cm_sw_r_cp_apa_i_w0XgFb3VEWAZ0
And this paint set later on when you feel comfortable in your skills. https://www.amazon.com/dp/B009162PWU/ref=cm_sw_r_cp_apa_i_-0XgFbMRFMBNZ
With the above resources you'll be able the paint the starters with ease as you'll have 6 minis to learn with and with the two starter sets you'll have some brushes and paints. Then expand your paints the final paint set.ane get the shade when you want to start washes and watch a few videos on that and dry brushing.and you'll be amazed at what you'll produce in a short amount of time.
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.