DM screens:
Wet-erase battle mat and markers:
A pound of dice:
Tiles are more fun for some than dry-erase.
Also fun are lots and lots of dice.
The Chessex mats are really awesome.
For playing around with game design, I'd get a Pound-O-Dice. Being in a variety of types and color makes them easy to use when you want to test different things.
I'd also get a big pack of college notebooks and a bunch of pencils and pens in various colors. Computers and printers are great, but I find the tactile feel of holding tools is good to put you in the mindset of making tactile games like tabletop games are.
Also, I have two more suggestions that I think can never be overstated:
Play more games. Both because it's fun and to understand their designs. Play bad games and figure out why you find them bad. Try to find a tabletop game club or something; as collecting games is expensive (and geekdom shouldn't be about what we own), meeting other nerds and sharing games is a great way to enjoy a variety of games on a limited budget.
Your first game won't be good and won't be published or enjoyed by strangers. The same is likely true for your second and third too. Don't worry about it; we learn through mistakes. Try not to get caught up in trying to make the next Game of the Year as your first project.
Pound o dice
Hundred dice
140 polyhedral dice
100 assorted meeples
200 cubes
https://www.amazon.com/gp/aw/d/B006L935AQ?psc=1&smid=A18KUEAGS3OGVI
250 chips
Mexican picture cards
Canvas pouches
https://www.amazon.com/gp/aw/d/B017UYSOJC?psc=1
Hex grid board