Welcome to JREF!
It's extremely hot here in Tokyo this summer, they say that it's the hottest summer ever in the past 113 years!
I do not know how long you're staying - just visiting or living for a long term - in any case I hope you'll survive in this heat!
(1) where to get affordable groceries in the Jiyugaoka area (that's where I'm at for now).
Have you tried 大丸ピーコック (Daimaru-Peacock), a supermarket not very far from Jiyugaoka station? Or 東急ストア (Tokyu Store), a supermarket within the station?
自由が丘 スーパー - Google 検索
There are several supermarkets but some of them (Kinokuniya, Queens Isetan, The Garden) tend to be pricey - because Jiyugaoka is considered rather a posh residential area. Instead, these supermarkets are handling many imported food which are not so commonly available in every supermarket.
(2) Also, I would like to go hiking --- where can I go?
How about Todoroki Keikoku?
Todoroki Valley
It's not very far from Todoroki Station, and Todoroki Station is only 3 stops away from Jiyugaoka by Tokyu Oimachi Line.
It is not hiking but strolling to Kuhonbutsu (an old temple famous for huge nine buddha statues). Kuhonbutsu is the next station to Jiyugaoka.
There are many trees and greens within the temple and it's a nice walk.
If you want more real hiking, I recommend Mt. Mitake and Mt. Takao.
Outlying Areas of Tokyo
(3) And, lastly, I would like to eat some katsudon, where do they serve it good either in Shibuya, Meguro-ku, or Tokyo?
There are many restaurants in Tokyo specialized in Tonkatsu, and most of the serve katsudon/katsuju (tonkatsu cooked with egg and served on rice. Katsudon in a porceline bowl and katsuju in a square japanese box.)
One of the easiest to find is まい泉 (Maisen) in Tokyu department store adjacent to Shibuya station.
The 9th floor is full of restaurants, and you'll find Maisen here.
渋谷駅・東急東横店は、2020年3月31日(火)をもって営業を終了いたしました。 |東急百貨店公式ホームページ
"Kurobuta katsudon" is JPY1,420.
Business hours 11:00 - 22:30