I am learning how to analyse NGS data. I have a data for 192 samples. These were obtained through a targeted sequencing library prep.

I have 192 samples, but technically I have received 2 files for each sample. For example:

 - sample1_TTGCCTT_L008_R1_001.fastq.gz  
 - sample1_TTGCCTT_L008_R2_001.fastq.gz

Presumably the reason there are 2 files is because paired-end sequencing was performed. I've been reading around on the various steps of NGS analysis but I can't seem to find an answer to the following question:

How to handle paired-end reads? Do you do have to merge them and if so when? Before alignment presumably? Also, do I have to uncompress the fastq.gz before I do anything? I am very new to NGS so apologies if these are really basic questions. Thanks.
You don't need to merge the R1/R2 reads. You don't say what kind of data this is but generally if you are aligning to a reference then you would use the two files together with an NGS aligner. Since the files contain reads from the same fragment their alignment to a reference provides spatial information. 

All extant NGS tools should understand gzipped files. You should not need to decompress then during analysis (note: there may be some exceptions depending on very specific programs).

(An aside: If the reads are longer than the 1/2 size of the insert then they can overlap in the middle. )

    Reads will overlap in this case
    
    |------------------------------>100 bp|    R1 - 150 bp
    |-------------------------------------|    Fragment 250 bp
    |100 bp<------------------------------|    R2 - 150 bp
     
    and will not here
     
    |-------->                            |    R1 - 100 bp
    |-------------------------------------|    Fragment 350 bp
    |                           <---------|    R2 - 100 bp

