What is the shape of SF4 (including bond angles)?

Within the context of VSEPR theory, you can count electrons to determine the electron geometry (“parent” geometry).
Sulfur: 6 valence electronsFluorine: 7×4 valence electronsTotal: 34 valence electrons
You can put sulfur in the middle because fluorine tends to make single bonds. Therefore, you can put 6×4 on each fluorine, 2×4 to account for four single bonds, and 2 for the last 2 valence electrons available.
As a result, you have 5 electron groups, so the electron geometry would be trigonal bipyramidal. With one lone pair of valence electrons, you get a seesaw molecular geometry.
Note though that the structure is distorted a bit due to the repulsive forces of the lone pair of electrons you see (not bonded).
So, that bends the axial fluorines together a bit. Normally the axial ##”F”-“S”-“F”## angle would be ##180^@## (because of course they are exactly opposite/antiparallel to each other), but it becomes ##173.1^@##, and normally the equatorial ##”F”-“S”-“F”## angles are ##120^@## (because the horizontal plane would distribute three connected atoms evenly over ##360^@##), but they become ##101.6^@##.