How much cash/T-Money Card?

Our Itinery looks like this: Duration of the Trip 4 weeks, 4 days on Jeju. Main Airbnb located in Seoul.

Since our Airbnb is located within Seoul I've already decided that I'm also getting a climate card (but open to other recommendations) cause we will probably do a lot of moving around within Seoul and it seems like a good deal for the span of our stay. I read though that as soon as we get out of Seoul a T-Money Card is recommended. On top of that apparently a lot of Foodstalls and such only accept Cash.

Since this is my first in Korea or Asia in general I am a little lost on how much Cash I should take out and how much I should load on the T-Money Card. I am a big Foodie so I am planning on visiting a lot of Street food places and I am planning on shopping quite a bit, also for little souveniers in those corner shops.

Any other tips regarding our Trip are very much welcomed and if you need more details on the Trip id be happy to give more Info!

Also, I'm hoping this isn't my last trip to korea so i'm not too worried about having a bit of cash or t-money credit left over at the end of our trip.