According to liberal ideas, what kind of society should we aim for? How would such a society make individuals better off?

Libertarianism is the view that anyone may engage in any act he wants, provided, only, that he does so with his own legitimately owned property, and does not violate the non-aggression principle (NAP). How do you get to own property initially? You do so by homesteading virgin territory; mixing your labor with it. How may property be transferred afterward? In any way that does not violate the NAP: trade, barter, lending, gambling, buying, selling, where both parties agree, and there is no fraud. What is the NAP? Keep your mitts off of other people and their property, unless invited to do so, and do not threaten initiatory violence.

What role does the market play in such a society? How different is this role from the role that other political and economic doctrines designate to the market?

The market is but one institution compatible with libertarianism. Others include gift-giving, charity, communalism (where people agree to live in a commune, a kibbutz).