A G-delta-sigma that is not F-sigma?

You want a $\Sigma^0_3$ set which is not $\Pi^0_3$. The canonical answer is a "universal $\Sigma^0_3$ set". You can find these concepts in books on Descriptive Set Theory (such as the one by Moschovakis or the one by Kechris).

A specific example: Let $N_{10}$ be the set of normal numbers (each digit appears with the right asymptotic frequency in the decimal expansion). Then $N_{10}$ is a $\Pi^0_3$ set which as complicated as $\Pi^0_3$ sets get (in particular: every $\Pi^0_3$ set is a continuous preimage of it). In particular, it is not $\Sigma^0_3$. So the complement of $N_{10}$ is what you want.

References:

1. Haseo Ki and Tom Linton, "Normal numbers and subsets of $\mathbb N$ with given densities", Fundamenta Math (1994). MR1273694

2. Goldstern, "Complexity of uniform distribution", Mathematica Slovaca (1994). MR1338422

Any dense $G_{\delta}$ with empty interior is of II Baire category, and cannot be $F_\sigma$ by the Baire theorem (and of course it is in particular a $G_{\delta\sigma}$).

Here are some examples from recursion theory which are boldface $\mathbf{\Pi^0_3}$ (the first is $\Pi^0_3(\emptyset')$ and the others are lightface $\Pi^0_3$) but not boldface $\mathbf{\Sigma^0_3}$:

The collection of weakly-2-random reals;

The collection of Schnorr random reals;

The collection of computably random reals.

References:

1. The Arithmetical Complexity of Dimension and Randomness. John M. Hitchcock, Jack H. Lutz, and Sebastiaan A. Terwijn. ACM Transactions on Computational Logic, 2007.

2. Descriptive set theoretical complexity of randomness notions. Liang Yu. To appear.