I've been vegan for about nine months, and was vegetarian for a year and a half before that. It's difficult at first, but you get used to it after a while, and making the transition from vegetarianism probably made it easier for me. The major difference is in the mentality and thought process Vegetarianism is a diet; veganism is a lifestyle. Here are some tips and advice I would offer to new vegans:
Check the nutrition labels on every thing. Obviously, you can't eat meat and you can't eat anything that contains milk or eggs. Watch out for less-obvious animal-derived ingredients like glycerin (made using animal fat), gelatin (usually made from ground horse bones) and casein (milk protein). Lists like
these are your friends when you see an ingredient you're unsure of.
Honey is an area of dispute, but most vegans don't eat it.
Doing research on what you buy helps in general. For example, most normal sugar isn't vegan. There are two major processes used to refine white sugar, and one of them uses animal bone char. Unfortunately, most companies use both and mix the two in their product. When looking for vegan sugar, look for "evaporated cane juice" in the ingredients. It isn't a fancy name for sugar; it refers to the way in which it was processed. Evaporated cane juice and most sugar marketed as organic will likely be vegan.
As a side note, many wines actually aren't vegan because of animal products used in making it.
If you go shopping for clothes, be cautious of leather (obviously) and fabrics like wool, cashmere, etc. If you're buying lotions or cosmetics, be careful of ingredients like lanolin.
While you don't have to shop at specialty stores or higher-end grocery stores, you're going to have an easier time if you do. More items are explicitly labeled vegan at places like Trader Joe's and Whole Foods, and there is even an aisle at my Whole Foods that's dedicated to specialty diets like gluten-free and veganism.
Be prepared to turn down most offers of food at places like school or the office (even things like gum). Assume that everything is non-vegan until you can read the ingredients, especially baked goods. Eating at normal restaurants is a hassle, and I honestly try to avoid it if at all possible. Try to search for vegan restaurants in your area, scout a normal restaurant's menu online before you go or simply eat beforehand.
There are plenty of substitutions for vegans. For milk, you can use soy milk, rice milk or coconut milk as a substitute. They even have non-dairy ice cream! For meat, look for faux-meat without milk or simply use tofu. I honestly don't really like faux-meat, because the taste is too close to normal meat.

Cheese is really the only thing veganism doesn't have a good answer for, in my opinion. Most of the vegan "cheese" available doesn't taste that good, and the texture is normally noticeably different than normal cheese.
Please, please, please take supplements. They will make your life much easier and prevent any possible nutrient deficiencies. I recommend using the following supplements every day: a multivitamin with B12 (I take Men's One A Day), an iron supplement, possibly a calcium supplement and possibly a Vitamin D supplement. Calcium and Vitamin D supplements may not be necessary depending on how much you get from your diet, but iron and Vitamin B12 supplements are almost required for long-term veganism. Also be sure that your supplements do not contain gelatin, since many manufacturers use that in their supplements.
Lastly, be prepared to encounter ignorance and questions about your veganism. Jokes like the one above aren't uncommon, and unfortunately most people are ignorant to how all-encompassing veganism is in life. You have to investigate what you eat and drink all day, every day. Even the clothes you buy and wear It truly affects almost every aspect of your life, but it can also be very rewarding. Good luck!