There are a number of kitchen gadgets that any cook needs in the kitchen, such a high quality knives and a cutting board or seven. However, there are some kitchen tools that you don't regularly find in a vegan kitchen (e.g., a meat thermometer) and others that you'll find more often. Below is a list of kitchen tools that I find indispensable in my kitchen. Since I can't prioritize, I put them in order of least expensive to most expensive.
Veggie Peeler ($2-15)
This should go without saying, but I'll say it anyway. Any kitchen that sees a lot of veggies should have at least one veggie peeler. To be honest, you should have more than one (unless you do your dishes or run your dishwasher daily). We have two and it's a rare week when they both aren't used. You can go old school and get one of the basic metal ones, upgrade for a softer handle grip (my preference), choose the Y-shaped model, or even go fancy and get one that has one ridged edge.
Measuring Cups/Spoons ($5-20)
This is another obvious addition in any kitchen, not only a veg*n kitchen. The key here is variety. I have the following in my kitchen: two sets of dry measuring cups that include some combination of 1/4, 1/3, 1/2, and 3/4, and 1 c. measures; two 1 c. liquid measures (pryex and one of the oxo easy-read plastic ones); three 2 c. liquid measures (two pyrex, one oxo); a 3 c. liquid; a 6 c. liquid; an 8 c. liquid; and three sets of measuring spoons. I certainly use some more than others, but occasionally you need to measure 1/16 tsp. of something or another, which is when the variety comes in handy.
Garlic Press ($5-20)
Shana and I joke that onion and garlic are their own food group in our house, but we're kind of serious. This is another kitchen tool that any kitchen should have. In fact, get more than one (I have two)! Although you can certainly mince garlic by hand or use something like a small grater, a garlic press will save you time and add increase the garlic flavor because you're bursting the cells of the cloves. The only really important thing I think all garlic presses should have is a little plastic or metal bumpy piece to push back through the holes; it makes cleaning these so much easier. Finally, with a garlic press, a recipe with a ton of garlic, such as garlic wine pasta is a breeze!
Jars(s) with Tight-Fitting Lids ($??)
Having a jar or five with tight-fitting lids is a magical help in the kitchen. I have a at least five of varying sizes and I use them all the time. Anytime I make egg replacer, it's in a jar with a lid. I always have a batch of Worcestershire sauce in the fridge and it's a jar with a lid. My overnight oats go in my lidded jars. When I need to make a flour/water mixture to thicken a dish, such as in wild rice soup or greek pitas, I mix the flour and water in a jar with a tight-fitting lid. Have I convinced you yet?!
Gigantic Bowl ($??)
I have a variety of cooking (distinct from eating) bowls in my kitchen. While I use them all occasionally, the bowls I use the most are the giant ones. I'll use them to hold the shredded cabbage for moo shu or when I'm making a batch of zucchini bread or when I need a lot of room to get my hands messy, like in chickpea cutlets. My three biggest bowl are also different materials. I have a large ceramic bowl (least used), a large metal bowl, and a large plastic bowl.
Salad Spinner ($10-30)
These things make washing leafy greens - including fresh herbs - easy and fun! Truth be told, I have two of these as well. I have a large salad spinner for when I'm dealing with large quantities of leafy greens or making salad for a dinner party. But I also have a smaller sized salad spinner. This smaller size is perfect for making a large meal-sized salad for myself or for serving salad with another item for both of us.
Rice Cooker ($20-75)
Rice shouldn't be hard to cook on the stove. I'm sure I'd get better if I practiced. However, I'm never practicing because my rice cooker is amazing. All you have to do is add rice (or other grains) and water, press the button, and you have rice for dinner. Having a rice cooker is great when rice is an integral component of dinner, like in broccoli rice bake or fried rice, but also when it's an accompaniment, like under BBQ tofu. Rice cookers come in a variety of sizes, so you can pick one that suits your rice-eating habits.
Classically used in the U.S. for tenderizing tougher cuts of meat, a slow cooker (or Crock Pot) is an ideal things to have in a veg*n kitchen, too. Many vegetables cook extremely well in a slow cooker, and they can also be invaluable for cooking dried beans. Slow cookers can clearly be used for more savory dishes, such as white bean & kale soup or minestrone, but they can also be used for sweet dishes like applesauce. A slow cooker is great in the summer, as it doesn't heat the kitchen, and great in the winter because of all the warm dishes you can make. Slow cookers come in a variety of sizes and shapes and complexity, but for most families a 5-qt model will be the right size.
Many people think a wok is only good for cooking Asian style food, like my Asian greens stir fry. While a wok is good for that, it's good for all sorts of other things as well! I use my wok to make soyburger helper and garbanzo pasta. Woks are widely variable in price, as a lot depends on what they are made of and if they have any coating. My wok is a carbon steel wok, which will eventually develop a completely non-stick patina without the use of a non-stick coating.
Immersion Blender ($30-150)
Also called a "stick" blender, an immersion blender is a great investment. Like a regular blender, these can be used for things like smoothies. However, they are also a bit easier to use when you'd like to blend anything hot. For example transferring hot soup to a blender can be difficult and dangerous. I use my immersion blender for savory soups, like my butternut squash soup, and also for sauces (vegan vodka sauce) and gravies (with herb-crusted tofu). Keep in mind that immersion blenders aren't quite a strong as a counter top blender, so they don't get things quite as smooth.
Take a look at the sauce pans in your kitchen. See how they have a flat bottom and then have essentially 90-degree angles with the sides? That's great for some things, but awful for sauces. If you plan to make sauces or gravies or roux even occasionally, a saucier may be a good investment. A saucier is a pan that doesn't have those 90-degree angles so that a whisk can get to every part of the pan. I can't tell you how many different ways I use mine, but it's necessary for the roux in potato soup and for the gravy in tofu pot pie.
Food Processor ($90-250)
This happens to be my most recent major kitchen purchase. I'm not kidding when I say it changed my life. I bought it to make my own falafel at home (for which it's awesome), but it's also been incredibly handy in a number of other dishes. I wouldn't attempt to make moo shu without it. I also use my food processor to make pesto and hummus. If you choose to purchase one, make sure yours comes with shredding and slicing blades, as those make your food processor even more useful.
Stand Mixer ($250-500)
My stand mixer has a place of honor in my kitchen. It sits where everyone can see it. It's beautiful and functional, which enhances its beauty in my eyes. I use my stand mixer for multiple applications. It works well for savory recipes like homemade seitan. But it is most useful (for me) for the sweet dishes. Nearly every cookie I make goes through my stand mixer: sugar cookies, chocolate chip cookies, and even the pumpkin cookies with maple glaze. *drool*