Having written about Pinterest recently I decided to list my top five places to do costuming research (not on the internet).
You remember the Library right? It is normally that large old-ish looking building on a college campus or in your town that holds books and other reference materials. Libraries are a great resource because there are also people there to help you find what you are looking for. They have tools like scanners, copiers, rentals, and nice quiet rooms for you to scan the books you find. It is amazing!
But seriously, many libraries have archives of fashion histories, magazines, ads, etc. that can greatly aid in your costuming research. I suggest taking time to read through the books and scan pages you want to references. This way you do not have to lug the books home and you save trees by scanning the documents to your thumbdrive or emailing them to yourself.
Much like a library, a bookstore has many different printed materials. Unlike a library one must buy these books before taking them home or scanning them. If you find costuming books that have many different eras in them I highly suggest purchasing one or two to keep in your own library at home.
Some bookstores will even special order a book for you if you request it!
There are many different kinds of museums. Some are more helpful than others. It just depends on what kind of costume design you are doing. For instance, an Air and Space Museum would not be good for a traditional concept of Pride and Prejudice, but it may be a great place to gather information for A Piece of My Heart (maybe).
Museums often have what most scholars refer to as "Primary sources". These include, diaries of people who lived during the time, original fashion of the time, photographs (assuming it is an era that came after the camera), etc. These things can be very useful when designing.
I like malls and thrift stores for many reasons. First, you can people watch. If you are doing a modern show this allows you to see how people are wearing today's fashions in an everyday setting. Secondly, you can see how fashion designers intended their closes to be worn (mannequins, photo shoots in stores, etc).
It also allows you to grab a few articles if you are currently pulling clothes for a show. (I like thrift stores because you can often find the exact same thing you wanted at the mall but for a severely reduced price)
Now I know this isn't exactly a place, but it does make for a lovely research area. There are many different places that do many different things in the clothing area. Traveling to these areas and speaking with locals or experts will help you gain insight to that area's different clothing customs. You also get a trip out of the deal and could potentially return home with authentic costume pieces to add to your design!