I have a bunch of Japanese skincare right now (and some on a wishlist). I've omitted the stuff that's hard to get in local stores. My skin type is sensitive-dehydrated, normal to dry in winter can get slightly combo in summer. I avoid mineral oil, fragrance, alcohol, parabens (for daily products), and silicones as much as possible.
-Muji Sensitive Cleansing oil - much prefer this to the DHC (avoid the mild version)
-Muji Sensitive Skin Toner High Moisture - lovely hydrating toner
-Muji Bath Salts are great too, I like the lemon grass.
-DHC Mascara - indestructible tubing mascara, the Isehan Kiss Me Heroine ones are also supposed to be v. good
-DHC Lip balm
-Gekka All in One Sleeping pack
-Cinderella Time Sleeping Pack Gel (don't know how easy this will be to find in stores)
-Kohaku-Hada Amber Moisture Lotion - This is an alternative to the Hada -Labo Gokujyun HA lotion that everyone loves but I personally prefer this one. The amber is supposedly anti-ageing.
-Minon Amino Moist masks (though a lot from this line looks interesting for sensitive aging skin)
-Lululun Prestige Red masks
-Shiseido, Koh Gen Do, or Muji cotton squares
-Melano CC spot treatment - Vit C that I use all over the face, v. affordable and effective for my purposes.
-Stock up on sunscreen it is so much more pleasant to wear Japanese SPF, my favourite is Kose Sekkisei Sun Protection Essence Milk but it's pricier than everyone's alcohol laden favourite Biore Essence. Also like the Canmake Mermaid UV Gel and the Mommy! UV Mild Gel.
-More high end, I'm very impressed with the IPSA time reset aqua which is sort of a hydrating toner/firs essence.
-If you're into makeup, definitely look into Suqqu. Brushes from them and Chikuhodo are particularly fab but £££. Brow pencil from Shu Uemura is also great (unless you're blonde).
-Tsubaki(Camelia flower) oil for hair
I'm probably forgetting some... anyhow the Cosme Japan rankings are a great guide for exploring what's highly regarded (switch on autotranslate) and Ratzilla's blog is also a great resource. If you find an a friendly English speaking person in a shop just ask them what the best things are right now, they're very into #1 products and helpful.