Eat a balanced diet of fruit, vegetables (including some green veg), grains/carby foods (bread/rice/pasta) and some legumes (beans, lentils, chick peas, soy products, peas).
The legumes will give you complete proteins. A varied and balanced diet along the lines of above would give you most of the nutrients you need.
Take B12 (e.g. from a supplement), and get ensure you get sources of calcium and iodine (see link below) as you eliminate milk and cheese. There is nothing else in milk and cheese that you need.
Also, consider eating some nuts as you transition because you are reducing the amount of fat in your diet by eliminating milk and cheese and nuts can offset that and may be helpful for your body to adjust. Although it's also possible that this isn't strictly necessary. Consider almonds (calcium) and walnuts (Omega 3).
For a very short article on vegan nutrition I wrote see here
https://whytryveg.wordpress.com/2017/03/25/vegan-nutrition/ It's based mainly on scientific studies and opinions of professional nutritionists.
Consider to very slightly increase your snacks or portion sizes as you eliminate cheese and milk, to maintain calorie levels. Or, if you want to lose weight, don't bother.