Nagai Shinai
Workman
A great (and cheap) place to shop for winter clothes. Workman also carries a wide variety of boots and "work clothes".
York Benimaru
The largest (and most expensive) grocery store in Nagai. Komeri has an excellent selection and very good English sign-posting throughout the store. In some ways, shopping their feels like walking through a store like Trader Joes in the US.
Komeri
A quite reliable home goods store. Komeri has everything from office supplies, to plants, to tools, to kitchen supplies, to furniture and bedding.
Musashi
A slightly smaller and cheaper home goods store than Komeri. The quality and selection here is still excellent, however, and rangers from potted plants to bathtub goods, blankets, lightbulbs, rugs, etc.
Yamazawa
York Benimaru's budget counterpart. Yamazawa doesn't have quite the luxury feeling or large selection of York, but, the prices are better and it still has all the basics you could ask for. A solid grocery store. Note - attached to York is also a pharmacy, a dry-cleaners, a Yamagata bank ATM, and often some tiny little food stalls.
Yamada Denki
Yamada Denki is an electronics store chain that can be found all over Japan. All the basics are here - dishwashers, fridges, vacuums, kitchen electronics, computer monitors, USBs, camera parts, phones, laptops, etc. The supply isn't huge, but, you should be able to find most basic stuff here (there are even Kotatsu tables and Nintendo games!).
K's Denki
K's Denki (literally, K's electric) is another electronics shop. The supply is similar to Yamada Denki, with perhaps a slightly different offering (more electric massage appliances, for example).
Fashion Center Shimamura
A small but surprisingly well-stocked clothing store. You can buy work clothes, pajamas, workout clothes, etc.. Both men and women's clothing. Prices are more than fair.
Kawachiyakuhin Nagaiten
One of a few pharmacies in Nagai (there are actually quite a number). Quite a large supply of medical goods, and a proper pharmacist staff on hand to help. Just don't expect English...
Outside Nagai
Uniqlo
The best place, hands down, to shop for winter clothes. Uniqlo has a large lineup of thermal wear and high quality jackets. In summer, you can still go to Uniqlo for some of the best quality clothes Japan has to offer.
Nitori
Nitori is something of a luxury or medium-high end home goods store. While the price is a bit higher than some other places like Komeri, the quality is really exceptional. This is definitely a place worth checking out to furnish your home-away-from-home.
Don Quixote
An...unusual place to shop at. DonQuixote is a colorful store that seems to sell a little bit of just about everything - clothes, food, electronics, shoes, coffee mugs, an XXX section that's carefully curtained off, DonQuixote has it all. A real Japanese experience.
Hardoff
A great store to find used goods of all sorts and varieties. The bottom floor has clothes, furniture, shoes, kitchen goods, sports goods, and even vintage art scrolls and decor. The top floor has electronics ranging from heaters to computer screens.
Bookoff
The sister store to Hardoff, Bookoff sells second-hand books, manga, and video games. There is even a small English section in the back. For Bibliophiles, definitely a recommended place.