This is your foundation, but just because it's your foundation doesn't mean you have to spend a ton on it (here are a bunch of options starting at $90). You do need to know what kind of mattress you have, because that dictates what sort of bed frame you need. Memory foam and inner spring (the traditional kind) mattresses can both work on a platform bed (the Floyd Bed is a stylish, affordable platform bed at $650 for a full/queen), while inner spring mattresses require a box spring on a bed frame without slats or a platform. That comes down to style preference, but I believe a platform bed actually makes for a more comfortable night's sleep since they're so firm and supportive.
However, bed style may not be the only factor that sways your decision; not using a box spring may void your mattress warranty.