I understand that finding higher homotopy groups of spaces is difficult because there's no equivalent to the van-kampen theorem for n>1 and so building homotopy groups can't be done easily by deconstructing a space in to simpler spaces which we know the homotopy groups of. I was wondering though if this could be done for special cases such as the one-point union (wedge product) of two spaces. I'm asking because I want to be able to find the homotopy groups of bouquets of n-spheres in terms of the homotopy groups on a single n-sphere; or explicitly if possible, but I know how hard it is to work out the homotopy groups of spheres so I'm not getting my hopes up.

One approach I've thought about is trying to find some nice fibrations from the bouquets to the single spheres and then using the long exact sequence of homotopy groups for a fibration to deduce some properties of the groups; however I'm pretty new to the concept of fibrations so I'm not sure how to go about constructing such a map or even if it's possible.

So I guess my question is 'Is there any relatively simple method for deducing properties about the wedge product of two spaces, specifically spheres'. Even any special cases like the 1-dimensional case would be useful for my work. Any help would be greatly appreciated.