How do you make pudding without lumps?

Pudding recipes do vary in the method in which they're cooked (some heat up the milk, whereas some combine everything then heat it up).

So, for recipes which involve heating the milk/cream, and adding it to a cold egg mixture and heating again: When it comes to adding the milk to the eggs, add a tiny amount first and whisk like mad until the pudding is smooth and combined. Then slowly add the rest of the milk, whisking constantly. Keep the pan on a low heat, whisking whilst the pudding cooks, remembering to scrape the sides and base of the pan to avoid burning. (Yes, you do need a whisk, a spoon or a fork will not suffice). And if all else fails, put it through a sieve at the end.

For recipes where everything is combined then heated up together: Firstly, make sure all your cold ingredients are thoroughly combined - use an electric hand whisk if you have one; this really gets rid of lumps and results in a very light pudding. Pour the combined mixture into a pan, medium heat (just until everything warms up). Keep whisking! Once the mixture has warmed, turn the heat down to low and whisk until it's thickened.

For both methods, stop cooking as soon as the mixture has thickened, otherwise you risk curdling the eggs (however if there's sufficient flour/cornflour in the recipe, this should not happen). It's also important to keep the heat low once the pudding is warm, but not yet thickened, since it tends to thicken quickly.