After a relocation to a new city one usually needs a lot of small things to buy. You hardly can bring every little item from your previous home. The first solution you might think of when you a need a jar or a basket is usually - IKEA.
IKEA. There are three big IKEAs around Moscow - all of them are on or slightly beyond MKAD road, so it is quite a trip. There are also mini IKEA in Afimall and Aviapark malls - much closer to the city center. They have a good selection in all categories, but its reasonable size is more pleasant and efficient compared to its huge siblings outside Moscow. It is a perfect stop if you need to buy some things for organizing or home textiles and kitchen utensils. If you know what you need you can make a quick visit and do not need to walk through the entire store (as is typically the case in a standard full-size IKEA store). Large furniture is also on display and can be ordered for home delivery. Твой дом (Tvoy Dom) literally means 'your home' and that is exactly what they offer: anything and everything for your home. From furniture, paints, pans, cutlery, washing machines, garden furniture to groceries. The quality of the food is usually good and they offer many international brands. Last but not least, Tvoy Dom offers an impressive selection of wines and beers! Also gluten free. They are reasonably priced and there are special offers on a regular basis.
There are four hypermarkets, mostly on MKAD road: https://tvoydom.ru/shops/
Jysk. A Danish chain of shops which sell kitchen and bathroom utensils, textile, linen, tableware as well as some furniture. Their shops are more easily reachable: four of them are in a walking distance from metro. Its atmosphere is much less hectic than in IKEA. Choose what you like on their website and go: https://jysk.ru/vannaya
Всё для дома (which means "Everything for your home") / Хозяйственный магазин ("Household items shop") - challenge Yandex.Map with any of those word combinations and you will see a lot of options, sometimes even very close to your place. These are not names of shops, but a type of one. Instruments, lamp bulbs, simple kitchen utensils, flower pots - this is what they normally sell. Sometimes cosmetics or porcelain stores got included into this category, so read the description before going.