We have carried out a search for optically visible post-asymptotic giant branch (post-AGB) candidates in the Small Magellanic Cloud (SMC). First, we used mid-IR observations from the Spitzer Space Telescope to select optically visible candidates with excess mid-IR flux and then we obtained low-resolution optical spectra for 801 of the candidates. After removing poor-quality spectra and contaminants, such as M-stars, C-stars, planetary nebulae, quasi-stellar objects and background galaxies, we ended up with a final sample of 63 high-probability post-AGB/RGB candidates of A-F spectral type. From the spectral observations, we estimated the stellar parameters: effective temperature (T_eff_), surface gravity (logg) and metallicity ([Fe/H]). We also estimated the reddening and deduced the luminosity using the stellar parameters combined with photometry. For the post-AGB/RGB candidates, we found that the metallicity distribution peaks at [Fe/H]~-1.00dex. Based on a luminosity criterion, 42 of these 63 sources were classified as post-red giant branch (post-RGB) candidates and the remaining 21 as post-AGB candidates. From the spectral energy distributions, we were able to infer that 6 of the 63 post-AGB/RGB candidates have a surrounding circumstellar shell suggesting that they are single stars, while 27 of the post-AGB/RGB candidates have a surrounding disc, suggesting that they lie in binary systems. For the remaining 30 post-AGB/RGB candidates the nature of the circumstellar environment was unclear. Variability is displayed by 38 of the 63 post-AGB/RGB candidates with the most common variability types being the Population II Cepheids (including RV-Tauri stars) and semiregular variables. This study has also revealed a new RV Tauri star in the SMC, J005107.19-734133.3, which shows signs of s-process enrichment. From the numbers of post-AGB/RGB stars in the SMC, we were able to estimate evolutionary rates. We find that the number of post-AGB and post-RGB candidates that we have identified are in good agreement with the stellar evolution models with some mass-loss in the post-AGB phase and a small amount of re-accretion in the lower luminosity post-RGB phase. This study also resulted in a new sample of 40 young stellar objects (YSOs) of A-F spectral type. The 40 YSO candidates for which we could estimate stellar parameters are luminous and of high mass (~3-10M_{sun}_). They lie on the cool side of the usually adopted birthline in the HR-diagram. This line separates visually obscured protostars from optically visible pre-main-sequence stars, meaning that our YSO candidates have become optically visible in the region of the HR diagram usually reserved for obscured protostars. Additionally, we also identified a group of 63 objects whose spectra are dominated by emission lines and in some cases, a significant UV continuum. These objects are very likely to be either hot post-AGB/RGB candidates or luminous YSOs.