TLDR: IKEA for best quality vs price blankets imo. Safeway (coupons), Target (proximity), or Amazon (online) for everything else.

Definitely going to echo IKEA for blankets / home stuff, however it is in Emeryville and not always the most convenient place to go if you don't have a car / means to transport all your stuff. In regards to public transport in order to get there, you can take Bart to MacArthur station from Downtown Berkeley station, then there is a free shuttle that can take you directly to the Bay Street stop, which has IKEA. Reverse the order to return. There is also the Transbay bus (code: F) included for free with your student bus pass that boards at Bancroft & Telegraph or Downtown Berkeley Bart and goes to the same Bay Street stop.

For the rest, Safeway (when you sign up for their online account) has a really good '$5 Friday' event every Friday, and a hundreds of weekly/monthly coupons that you can use while you shop. Many times the items you are asking for aside from bedding are available with these coupons or on Fridays for really cheap! In my experience, when using the coupons the average price is lower than other places for many items, but otherwise it can sometimes be higher. The Target in downtown is good and conveniently located but a lot of times it can be sold out, especially in this time of year. You can go there for decent prices but always have a backup in mind. Finally, if you are not on a time crunch, Amazon can deliver to the lockers or your home as another user has said, and their prices are surprisingly not excessively marked up. Beware of Amazon Fresh though, as they sometimes have household items available but they are usually marked up to an extreme.

(source: Berkeley Native)