Simulations indicate that the first Population III stars should appear in minihalos of mass M = 105-106 Msol at z ≈ 10-30. To assess the detectability of these objects, theoretical models of these stars and their surrounding HII regions are used. We assess the plausibility of detection with the upcoming James Webb Space Telescope (JWST), using the gravitational lensing provided by the galaxy cluster MACSJ0717.5+3745. The conclusion is that the detection of these objects is highly improbable but not impossible. To investigate the prospects of detecting and identifying the first galaxies, the spectral synthesis code Yggdrasil is introduced. According to this code, JWST may be able to detect Population III galaxies with stellar masses as low as 105 Msol at z ≈ 10 in unlensed fields. We find that, over limited redshift intervals, it could be possible to use Hubble Space Telescope (HST) and/or JWST broadband color criteria to single out Population III galaxy candidates. The prospects of detecting gravitationally lensed Population III galaxies with JWST and HST is investigated. A lower limit to detect ≈1 Population III galaxy of ε ≈ 10-2 (HST/CLASH) and ε ≈ 10-3 (JWST using MACS J0717.5+3745 as lens) is derived, where ε is the baryon fraction converted to Population III stars in a host halo. By fitting HST/CLASH data to Yggdrasil and comparison grids, two Population III galaxy candidates are discovered. These two candidates are the first Population III galaxy candidates discovered at z > 6.5. A highly-magnified and doubly lensed extremely high-redshift (z ≈ 7.8) object is also identified. Finally the prospects of detecting core-collapse (CC) supernovae (SN) from the first galaxies at z ≈ 5-12 are investigated. The prediction is that no primordial SN is detectable, but 2-3 CC SN should be discovered by the HST/CLASH.