Unfortunately, furniture, including boxes and chests is just for looks. You are supposed to be able to place books in bookcases but most of them are still bugged. All of my books are standing on tables.
In the mean time, you can place books scrolls and other items in your home but if you're feeling really cramped for space, you will need bags and boxes for your inventory and bank slots.
For inventory space on your toon, there are a lot of simple quests you can do for nice bags and if you gather and forage some basic tier one items, you can easily craft some elm strongboxes for your bank space.
In addition, you have an additional 2 slots called a home vault. You can access these from inside your apartment and can place any bag or box in each slot for even more storage.
Good luck and welcome to Norrath.