A simple bread pudding doesn’t have terribly many ingredients, which means, according to Joe’s Inverse Law of Ingredient Dynamics*, that the ingredients you use should be good ones. Good milk, fresh eggs and fresh, real vanilla beans are all desirable. However no single ingredient will effect the quality of your pudding like the bread. It needs to have just the right texture, so you want to be choosy. What are consequences of not picking the right bread? In a word: mush. Typical mass-market sandwich bread will simply not make good bread pudding.
Oh sure, you can use things like brioche, croissants or panettone if you want. However most of us don’t have that sort of stuff laying around in our kitchens getting stale. And anyway it’s called “bread” pudding, right? My feeling is that anyone should be able to make a great bread pudding with stuff that’s more or less on-hand.
So, the bread. You want something white, with a pretty tight crumb that’s also at least a little bit firm. If it’s also stale, so much the better, however stale Wonder bread is out. For all you readers in America, think Pepperidge Farm white sandwich bread, or an organic white sandwich loaf from a Whole Foods or local bakery (I bring up organic not because I advocate organic, but because the bread won’t have the same preservatives, so it’ll be drier and tougher than most big national brands). That sort of thing you can use right out of the bag, though at least some staling will make it even better.
But do the best you can. There really are no rules here, but if you can be choosy, do be.
*States that as the number of ingredients in a given recipe goes down, the relative quality of those ingredients should go up.