However, I know that this is not always possible, so I would say that you could have as many samples as you can do in a triplicate replica analysis, at least.
I would also recommend to do the amplification in triplicate to cover the most of the variation that the PCR bias could introduce. You can do it with the same annealing temperature or changing it in a +/- 2 ºC variation.
In my point of view, the best quality/no. of sequences/price ratio is Illumina MiSeq technology. You would get sequences of c. 600pb, I do not know if it is enough for your study
For illumine Miseq, it is the most commercial source for 16S sequencing. Generally people use V3-V4 region of 16S. However, you should look up some papers to check which region they use to identify human gut microbiota. Different region of 16S rRNA gene would have different resolutions.