Like anything else, sewing is much more enjoyable when you have the right equipment for the job.
Your sewing machine is the most important piece of sewing equipment you will ever own. Buying a new sewing machine can be a daunting task, but taking the time to research and explore your options is well worth it. A sewing machine that's not well made or suited to your sewing style will only leave you frustrated, while the perfect sewing machine might have you crying tears of joys and busting through your fabric stash at record speed.
You can find some great tips for narrowing down your sewing machine search here. And the FREE Craftsy mini-class Sewing Machine Feet from A to Z will help get you started on your machine.
A serger is not a required piece of sewing equipment, but many sewists find them extremely useful. A serger sews over the edge of the fabric, trimming off the excess fabric, so you're left with a nice, finished seam. Sergers are frequently used when sewing with knits. There's a learning curve involved in mastering the use of a serger, but sergers run at high speeds, so a serger might be a good investment of time and money if you frequently sew knit fabric.
Good lighting is essential in any area where you have your sewing equipment set up. Even if the room has good overhead lighting, you'll probably find it necessary to invest in task lighting as well. A full-spectrum task light will help you see all of the tiny details on your sewing (and thread a needle on the first try!), without straining your eyes.
Surprisingly, there's a lot of ironing involved in sewing. Wrinkle-free fabric is much easier to work with and will help you create garments with crisp, professional seams.
An iron is definitely a piece of sewing equipment to invest some cash in if you can. For a long time I was using the $5 iron I purchased when I went away to college and I couldn't believe the difference when I started using a Rowenta iron that cost, well, not $5.
These days you can also find cordless irons, which can be very handy. You'll need a proper ironing board, as well, but this doesn't have to cost a lot of money.
This is another item you could do without, but it's very nice to have. This three-dimensional model of a torso is useful when sewing so you can see the fit and drape of a garment and make any necessary fit adjustments. Dressforms are typically adjustable, so you can mimic your body's shape on the form for the best fit.
In addition to the sewing equipment listed above, you'll want to have basic sewing tools on hand.