There are a lot of reasons why you may want to whisk by hand besides lack of equipment. Maybe you want to keep things quiet and simple, or the baby is sleeping nearby, or you're just trying to be more unplugged in the kitchen. Maybe you just don't want to haul out your stand mixer or maybe your electricity is out and you still want to bake that cake. Or maybe whisking by hand is the only exercise you're going to get today! Whisking by hand also allows you to keep a close eye on what's happening so you can stop when your cream or whites are just perfect.
With both cream and egg whites, you should be sure your bowl is large enough to allow for vigorous whisking and the expansion of the whites or cream. You are incorporating air, so your whisk needs plenty of room to move around. A balloon whisk is nice in that it helps to incorporate in more air but it's not necessary.
If you have a copper bowl, then you are living la dolce vita! There is a chemical reaction between egg whites and copper that helps the whites to whip up and stiffen. But don't worry, a regular (but not plastic) bowl will work, too, just use a pinch of cream of tartar to help mimic the copper's influence.
Whatever you do, be sure your bowl and whisk are very clean and grease-free as fat, as well as bits of egg yolk, will hinder your progress.
If possible, let your egg whites come to room temperature before beating. You can hasten this by placing your bowl of egg whites into a bowl of hot water for a few minutes, stirring occasionally.
Begin whisking by vigorously swishing your whisk back and forth to break up the egg whites until they are foamy. Then start whisking in a more circular motion, lifting the whisk up and out of the egg whites. When to stop? That depends on how stiff you want your peaks. Visit Emma's visual guide for a look at what is meant by soft, firm, and stiff peaks.
Unlike with egg whites, you want your cream nice and cold when you whip it. If there's time, chill your bowl and whisk in the freezer for several minutes. Cream tends to splatter when you first start to whip it, so keep your strokes as small and contained as possible in the beginning. One the cream has started to thicken you can beat more vigorously. Add sweeteners and other flavors (such as vanilla) after the cream has reach soft peak stage (again, Emma's guide to peaks is here.)
Will your arm get tired? Possibly. If it does, just switch to your other side and do your best until your dominant hand is rested and then switch again. You can hold your bowl in one arm and whisk with another or you can whisk with your bowl on a surface in which case you may want to create a circular collar with a tea towel to hold the bowl in place.
It's true, there are times when I wish I had my hand or stand mixers back but usually I enjoy my mini workout with the whisk and bowl. As mentioned, I can keep a closer eye on things when whisking by hand and it's much quieter, too. And, I have to admit, whisking by hand makes me feel kind of tough and self-reliant. Who needs all those sissy electrical appliances when you've got a copper bowl and a forearm the size of a baseball bat?
Related: Quick Tip: Beating Egg Whites