Are some types of wines less likely to have oak than others?
Yes, and I hope you like white wine.
There's no consistent way to tell from a label. Often the small print on the back will refer to how the wine was fermented or aged, whether in stainless-steel tanks or oak barrels. Generally, light, fresh and fruity whites are your best bet. These include most sauvignon blancs, pinot grigios and virtually all rieslings (riesling and oak get along like New Democrats and Conservatives). But the list is long and tends also to include such wines as lean Rias Baixas from Spain, fruity muscat or moscato, and spicy gewurztraminer. Oak softens texture and adds flavours of its own, which can obscure the freshness of delicate whites. Most chardonnays, in contrast, are aged in oak, though there are exceptions, such as many Chablis and New World chardonnays that explicitly boast of their "unwooded" character on the front label.
The vast majority of reds spend time in barrel. But oakiness comes in degrees, both in terms of time maturation time as well as type of wood used. Many wines from the southern Rhône Valley, for example, are cellared in large, old (as opposed to new) vats, which impart very little, if any, discernible oak character. Côtes du Rhône is a good example.