Glycine increases main chain flexibility. Gly has a hydrogen in place of a side chain. With no side chain hindrance to rotation about the Cα atom the main chain can bend sharply at Gly residues. Gly also allows the chain to fit into tight places. It is precisely because Gly has no side chain that it tends to be highly conserved during evolution.
Alanine can be either on the inside or at the surface of a protein. Although the methyl group is nonpolar, it is too small to have a strict preference for hydrophobic environments. You may recall that ethanol is miscible with water but butanol is not. The frequency of Ala in proteins is high, whereas the frequency for Gly is fairly low; too many Gly residues would cause the main chain to become too flexible, making fast protein folding very slow, or even impossible.
Proline's side chain is rigid and often forces a sharp bend in the main chain. Although the cyclic side chain itself is nonpolar, the large dipole moment of the peptide bonds of the polypeptide chain counteracts any tendency of the Pro side chain to flee from water. Consequently, Gly, Ala, and Pro fall in the middle of most hydrophobicity scales (one popular scale ranks Pro as a hydrophilic amino acid). Proline occurs frequently in β hairpin turns.