How to blanch vegetables: 1. Clean and trim your vegetables. Cut them if necessary to similar size pieces. If using a vegetable mix, I recommend blanching the different vegetables separately, unless you can estimate the exact blanching time of each kind. I usually blanch vegetables together, but I know when exactly to add them to the pot during the process, and even then it is only an educated guess. Of course it is not the biggest problem in the world if one vegetable is not perfect, but some vegetables fall apart if you try to saute them, after they have been overcooked already.
2. Bring a large pot of water to a rolling boil and add the vegetables. Blanch for 2-3 minutes, or until the desired tenderness is achieved.
3. Strain the vegetables in a colander, and run cold water on them until the are completely cold. Alternatively, you can prepare an ice bath for them and chill them even faster. Usually restaurants prepare vegetables this way, and store them for later use. If you want to use the vegetables right away, you can even saute them right after straining them from the hot water, or toss those veggies with some butter and seasoning, and serve immediately.
If you are using frozen vegetables, you shouldn't thaw them before cooking. Put them right from the package in the pot with the boiling water, and blanch as usual. Frozen vegetables usually require a little bit less time to prepare, because the freezing has already softened them.
How to blanch potatoes: Blanching potatoes for fries is a completely different technique that is done in hot oil, not water. Cut the potatoes for fries and rinse them. Rinsing is to remove the starch from the surface of the fries. It does not matter if you peel the potatoes or not. After that, strain the potatoes completely in a colander. Heat vegetable oil in a deep fryer to 325 F, and drop the potatoes in batches that are not too big. Blanch the potatoes for about 5 minutes, enough for them to become soft, but not change their color too much. Strain the blanched potatoes well and keep them cool until needed. Blanching potatoes is a method usually employed by food service establishments. I don't find it necessary for home cooking.
How to blanch tomatoes: Blanching tomatoes is also something that is necessary for restaurants, but rarely done at home. Core the tomatoes and cut a tiny "x" on the tip. Dip each tomato in boiling water for 1 minute, remove and place them in an ice bath. After the tomatoes are cold, peel the skin and use it for whatever recipe you were using. Here is one recipe made with blanched tomato.
Of course, blanching is not suitable for every kind of vegetable. Zucchini can be blanched, but usually there is no point in doing that. Mushrooms are another example of vegetables that don't need blanching. Some people blanch vegetables before freezing them. I don't see much reason for that, aside from preserving the color of the vegetables.