How to add Spheres[] sequentially from the surface of preexisting spheres?

spheres = Sphere[#, 1] & /@ NestList[
  With[{rp = RandomPoint[Sphere[#, 1]]}, 2 rp - #] &, {0, 0, 0}, 5];

chained spheres

If you need the spheres to be self avoiding, then you could look into this question.

All spheres are oriented upward.

NestList as in flinty's answer is a bit overkill. You can simply use Accumulate like this:

Graphics3D@Sphere@Accumulate[2 RandomPoint[Sphere[], 5]]

Output

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