What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?
I think the list on Wikipedia is incomplete:
The Hodge conjecture implies the Lefschetz and Kunneth standard conjectures, as well as conjecture D (for singular cohomology) over fields of characteristic 0.
The Tate conjecture also implies Lefschetz, Kunneth, and conjecture D (for etale cohomology) over all fields (not just characteristic $p$).
The reason is pretty simple. All of these conjectures are about the existence of an algebraic cycle: For the Lefschetz conjecture, a cycle inducing the Lefschetz operator. For the Kunneth conjecture, a cycle inducing the projector. For conjecture D, a cycle pairing nontrivially with a fixed homologically nontrivial cycle. In each case, it is easy to check that there exists a cohomology class with the specified properties which is a Hodge class / Tate class. Hence assuming the Hodge conjecture / Tate conjecture, there is an algebraic cycle as well.
In terms of the reverse implication, as far as I know there is none.
The period and Hodge conjectures imply the standard conjectures. Yves andré (Une introduction aux motifs)