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Wellcome Supermarkets Linshen Branch
This chain of supermarkets has branches all over Taiwan. They have excellent fresh fruit and vegetables at a reasonable price. They carry food and other items of both the domestic market as well as from overseas. They also have small deli counters that offer meat and fish of good quality.
Their prices are reasonable, and they often have excellent deals. Because there are so many, there's usually one in your vicinity and generally they're not too busy. Signs in the shops are usually in Chinese, so it may be a good idea to learn the character for whatever you're looking for.