I’ll preface this by stating that being vegan has nothing to do with being fit. It might be easier to eat better if junk foods are off-limits due to not being vegan. Plants are also more nutrient dense and have lots of fiber, which is important for effective dieting. However, veganism is an ethical movement and has virtually nothing to do with health or fitness.
That being said, I’m glad you’re becoming one of us now!
In order to lose body fat, you need to be in a caloric deficit. A pound of fat is about 3500 calories, so to lose a pound of fat a week, you’d need to average about 500 calories of a deficit each day. You can google “calorie calculator”, enter your stats, and receive a fairly decent approximation of your TDEE.
Then, simply take that number and subtract 500 (or any other size of deficit), and you’ve found your daily caloric target.
There are two ways to create a deficit – exercise, and dieting. Exercising, including high-intensity activities like running, burn too few calories over too much time to be a sustainable and effective way to lose large amounts of fat. Furthermore, you should ideally be following a structured resistance training plan, and too much cardio will impede your ability to recover from the workouts (which, in turn, makes it harder to gain muscle mass, which ultimately makes it harder to keep the weight off long-term).
As such, the ideal way to create the deficit is dieting. Purchase a small food scale and use it to weigh your foods, and use an app like MyFitnessPal to track your caloric intake each day. It doesn’t have to be perfectly precise – you only need a ballpark estimate of the caloric content of your meals, especially for calorically dense foods like nuts, to ensure you’re roughly meeting your caloric goals each day. After a couple of weeks, if it gets too burdensome, you can always stop tracking and just eyeball it (though this is more prone to error, since most of us are very bad at eyeballing portion sizes, especially while hungry).
As for what you should eat, don’t worry about it too much. As long as you’re eating a variety of fruits, vegetables, greens, grains, etc. that you generally like, as well as a decent amount of protein (e.g., legumes, tofu, seitan, vegan meats).
People like to talk a lot about macros (fat, carbs, and protein, the three “types” of calories you can eat), but as a beginner with (presumably) a good amount of fat to lose, you don’t need to worry about the specifics very much to see excellent progress. Focus on eating a variety of healthy plant foods and executing your new plan. You can always optimize your diet and exercise later, but there isn’t much reason to worry about the details when you’re just getting started.
Also, you should aim to weigh yourself once every morning after using the restroom, so you can take the weekly averages to gauge your progress. You’ll probably instantly lose some water weight when you start your diet, so keep that in mind.
As for what workout routine you should be doing, I would defer to Jeremy Ethier and/or Jeff Nippard on YouTube for this. In short, you should find a workout program (free or paid) that lets you train all the main muscle groups twice a week, focusing on progressive overload (lifting more weight each week as you get stronger). This will let you build muscle mass, which helps you increase your base metabolic rate as well as look and feel fitter. If you can, work with a trainer or experienced friend at first to learn how to use a gym and perform everything with good form.
Feel free to respond with any questions.