There are the standard utilities — mortgage, electricity, heating, phone and broadband. If working from home, you can reclaim a percentage of these bills when filing your tax return. If your studio is away from your home then the full spend is tax deductible.
Then there are the workspace basics — desk, chair, computer, software, printer/scanner, ink, paper, a lamp, a bookcase (and books), shelves, sketchpads, pens, pencils, a good external mic, headphones, external hard drive. Be sure to keep your receipts for tax deduction.
Kelli Anderson’s stand-up IKEA desk hack.
A few things more specific to the profession — Adobe CC, font licensing, LiveSurface and other mockup resources, MailChimp, web hosting, and domain registration (I use Namecheap). A good camera, tripod, lighting rig, and backdrop will help you shoot print work for your portfolio.
Other expenses might include travel for meetings, postage for letters and packages, classes from sites like Skillshare and CreativeLive, an accountant (unless you file your own returns), but that mostly covers it.
Related, from the archives: Reflections on design self-employment.