The metric raises and lowers indices according to

g_{ij} V^{j} = V_{i}

Where the thing on the left hand side means a sum over j :

V_{i} = g_{i0} V^{0} + g_{i1} V^{1} + g_{i2} V^{2} + g_{i3} V^{3}

This idea generalises to two indices as in these examples:

g_{ij}F^{jk} = F^{k}_{i}

g_{kl}g_{ij}F^{jk} = g_{kl}F^{k}_{i}= F_{il}

That is, to lower both of the indices one applies the metric twice.

So when you have a contraction of two 2-index objects like this:

F_{ij}F^{ij} = F_{ij}g^{ik}g^{jl}F_{kl}

Explicitly, using the Minkowski metric and taking F_{ij} to be a matrix of 1s

F_{ij} =

`| 1 1 1 1 |`

| 1 1 1 1 |

| 1 1 1 1 |

| 1 1 1 1 |

F^{i}_{j} = g^{ia}F_{aj}

`| -1 -1 -1 -1 |`

| 1 1 1 1 |

| 1 1 1 1 |

| 1 1 1 1 |

F^{ij}= g^{ia}g^{jb}F_{ab}

`| 1 -1 -1 -1 |`

| -1 1 1 1 |

| -1 1 1 1 |

| -1 1 1 1 |

as you said.

Does that clear things up at all?