If (like in your example) you know the hierarchyid values you want to manipulate in advance, you can do it directly, e.g.:
-- Place 1st node between 2nd and 3rd
UPDATE yourTable SET node = CAST('/8/2.5/' AS hierarchyid) WHERE value = 36;
-- Move 3rd node to 1st
UPDATE yourTable SET node = CAST('/8/1/' AS hierarchyid) WHERE value = 34;
If you need to get your new hierarchyid values dynamically, take a look into the GetDescendant() and GetAncestor() functions. Using that, the example would look something like this:
DECLARE @Parent hierarchyid, @Child1 hierarchyid, @Child2 hierarchyid
-- Grab hierarchyids from 2nd and 3rd node
SELECT @Child1 = node FROM yourTable WHERE value = 38;
SELECT @Child2 = node FROM yourTable WHERE value = 34;
-- Get the parent hierarchyid
SELECT @Parent = @Child1.GetAncestor(1);
-- Update 1st node to end up between 2nd and 3rd
UPDATE yourTable SET node = @Parent.GetDescendant(@Child1, @Child2) WHERE value = 36;
-- Update 3rd node to end up before 2nd
UPDATE yourTable SET node = @Parent.GetDescendant(NULL, @Child1) WHERE value = 34;
Note that the hierarchyids do not stay the same in this example. At least one will end up using a 'fraction' for its position (e.g. '/8/2.5/' instead of '/8/3/').