How can I find the second cheapest spanning tree?
The second-smallest spanning tree differs from the minimum spanning tree by a single edge swap. That is, to get the second-smallest tree, you need to add one edge that's not already in the minimum spanning tree, and then remove the heaviest edge on the cycle that the added edge forms.

If you already have the minimum spanning tree, this can all be done in linear time. See e.g. V. King, "A simpler minimum spanning tree verification algorithm", Algorithmica 1997, doi:10.1007/BF02526037.