cheapest way to build a gazebo?

Cheapest by a long way is a kit, honestly. Once you price up posts, roof sheets and fixings, building from timber usually costs more than a steel-frame gazebo bought in an end of season sale. If you want to build one anyway, keep it simple: four treated posts, a flat or single-pitch roof (polycarbonate or felted boards), post spikes or concrete for the posts. The roof is where cost and difficulty jump, so skip a hipped or pitched design for a first go. And check the height rules where you live before you start, some places need permission above a certain size. What size are you after?