When Amelia and I went vegan, I did it cold tofurkey (hehe). I simply couldn't stand the thought of eating meat, dairy or eggs anymore and haven't since. If I have a meat sneak like Amelia did, I would probably throw up. I have a hard enough time walking through the meat section (aka morgue) at the grocery store. The thought of putting it in my mouth makes me physically ill.
Amelia has a hard time throwing away food, especially food that had its life brutally taken to be in our refrigerator. However, since I wouldn't eat it, we threw the meat out. Amelia did finish some of the dairy and eggs we had purchased, but we haven't bought any since.
The non-food items are a different story. I had a car with leather seats. I have leather shoes, leather belts, and a leather football that was given to me as an award years ago. We have health and beauty products that were tested on animals. Once you go vegan, you realize we use animals like toilet paper! It's shameful!
However, we bought those things (or were given those things) before our blinders were removed. Some people would argue that we should throw it all out and torch our cars, but that's very wasteful in our opinion.
Part of being vegan is environmental (destroying the environment also destroys animal habitat, air and water) so throwing perfectly good non-vegan things out and replacing them with vegan things not only creates trash, but also requires more resources to create the replacements.
Another part of being vegan is to do the least amount of harm. The damage has already been done for the things you already own so it's better to take care of them so they last as long as possible to honor the animal who had its life taken, as well as prevent the needless use of additional resources to create new items.
Most sensible vegans recommend using what you already have, but don't buy any more of it. That seems like reasonable, environmentally and financially friendly advice.