There are a lot of kitchen gadgets out there. Honestly, although neat, you don't need the mass majority of them. Contained in this list are the basic items I think every vegan should have to create great meals.
Cast Iron Skillet
Tired of your tofu sticking? Worried you aren't getting enough Iron in your diet? Don't have time to go to the gym and lift weights? A good cast iron skillet is the solution! Why people ever stopped using them for everyday use I'll never understand. They work best for browning, frying, camp cooking, and pretty much anything else you'd use a skillet for. They do require a bit of extra care to maintain the "seasoning" that keeps them non-stick, but trust me, it's worth it. Both Lodge and Le Creuset make great pre-seasoned options. Invest a few dollars in a good quality one, plus take the 30 extra seconds after washing, to oil the surface and you'll have it for a lifetime. You can also find them at second hand stores, if you don't mind the extra work of re-seasoning them.
Good Quality Kitchen Knives
Even if you only purchase a Chef's Knife and Paring Knife, please buy good quality blades. You'll save yourself time, effort, money in the long run, and yourself from possible trips to the emergency room. There are several great companies out there but, Wusthof, J.A. Henckels, and Kyocera, are my favorites. If you go with traditional stainless steel blades, be sure to purchase a sharpener by the same company. Their sharpeners are set to hone the edge to the specific angle of their knives. Using a sharpener from a different company will offset your blade. If you aren't interested in having to maintain your blades, Kyocera Ceramic Knives are a great option, as the non-porous material doesn't dull like steel. The big drawback of ceramic is that they're a bit delicate. Don't drop them in the sink or use them to open can lids, you'll break them.
How on earth I went so many years without a food processor in my kitchen, I'll never know. Its easily the hardest working small appliance in my house. I use it for sauces, dips, dough, chopping, mincing, slicing, and blending. Again this is something you want to invest some money in. You really can't go wrong with a Cuisinart, considering they introduced the world to the food processor. They have several size and function options to choose from. My personal favorite is their Prep 11 Plus, its basic, but it has everything you need. Kitchenaid also has several good options. I recently got to demo their new 13 Cup with ExactSlice, and if you plan to do a lot of slicing, this is the one to go with. It works amazing and you can control the slice thickness with an exterior switch instead of the hassle of taking it apart.
A good stock pot is a wonderful thing. For everyday use an 8 quart is the perfect size. It's just right for soups, pasta sauce, rice, mashed potatoes, etc... Look for a pot that has a heavy bottom, as this will cut down on any sticking. I like Calphalon's Simply Calphalon 8 qt. Multi-purpose Stockpot, as it comes with a steamer basket and pasta insert for a reasonable price. If you'd like to get into canning, look for a 20 quart pot, like this one from Nordic Ware.
Want perfectly steamed veggies, dumplings, or a great way to keep tortillas warm for burritos? Grab a 10 inch Bamboo Steamer from Joyce Chen. They're inexpensive and seriously come in handy. I use mine for steaming homemade seitan, savory and sweet dumplings, and for quick and easy steamed veggies on nights when I'm not in a cooking mood.
I also suggest some heavy duty baking sheets and pans, a silicone collapsible strainer, a counter top compost bin, and a good quality blender. You'll find you have little use for much else and don't have to replace things as often, if at all, when you invest in a few quality items.