Veganism aims to eliminate the use of all animal products as much as is practicable. Animals feel pain and don't want to be killed-- this is the most basic driving instict of all living creatures. Of course animals kill each other in the wild for food, but we are humans and we can use logic and reason and we can think about morality. Morality is a complex philosophical subject, but I think few would disagree that needless suffering is bad. If we can satisfy our needs to live (clothing, food, shelter, medicine, etc), why would we make others suffer? Maybe you'd want to make a cow suffer for a cheeseburger, but you don't need cheeseburgers to live a full, healthy life. If you can eat something plant-based and get the same satisfaction, is there any difference in what you ate besides the suffering done to that cow?
Milk is heavily entwined in the beef industry. In order to produce milk, cows must be continually impregnated (by farmers), and they continually give birth. Their male calves cannot produce milk and will be sent to slaughter. Their female calves can produce milk, so they will meet the same fate as their mother: impregnation and milk production until they are "spent" and must too be slaughtered. Eggs are a similar story. Hens continually lay eggs and their male chicks are worthless (they don't have desirable characteristics for meat and cannot lay eggs) so they are immediately destroyed.
All animal biproducts inherently cause some suffering because they commodify animals and their bodies. We breed these animals only for their capacity to produce as much product as possible to maximize profits under capitalism. All other needs are put second to the bottom line.
The mainstream image of a vegan is a rich white woman who has access to specialty items. This image, while it does exist, is a harmful stereotype. Veganism needs to be for everyone and accessible to anyone. The issues around animal agriculture disproportionately affect minority groups and the working class. When giant pig farms dump their waste and contaminate the water supply, the surrounding communities suffer. Slaughterhouse workers must endure unbelievable trauma and danger every day at work and may fear for their lives if they want to speak up or quit if they aren't legal citizens. Fast food targets poor and minority communities, and thanks to government subsidies, can make unhealthy animal products as cheap and widely available as possible.
Vegan food does not need to be specialty and it does not need to be expensive. It's no coincidence that so many cultures around the world use plant-based recipes as their staple foods. These foods are usually the cheapest to produce and yield the most energy. And they are usually very delicious too!
Animal agriculture is a huge source of greenhouse gasses which contribute to climate change. It has been compared to the emissions from the entire transportation sector. I'd encourage you to watch the documentary Cowspiracy for more info about this. Animal agriculture is inherently more taxing on resources than plants because animals must be fed food and water to survive to slaughtering age. They do not convert all food and water fed to them into meat/product in a 1:1 ratio, so it is not efficient (plus, that food and water must be shipped to them, sometimes from far away).