Keep them separated, like Nicolas said, most modern NGS software should handle paired-end reads. One they're aligned, you should have a single SAM/BAM file containing reads from both ends.
Depending on your purpose, you may need to choose different tools. For variant calling using whole-genome-sequencing data, I used
bwa mem for aligning the reads. A good resource would be the Broad's Best Practices Guideline, which would cover the alignment step (note what version the the Guideline you're using; I've used the one for GATK 3.0, and they recently updated to 4.0 so I can't comment on the latest one.)
For RNA-Seq, if you have Illumina short reads, you probably want a splice-aware aligner in order to detect cases like a read spanning an exon and part of a retained intron. I like STAR personally, and HISAT2 is also popular and a bit more recent one.
And .gzipped files are often supported; with STAR you simply specify
My RNA-Seq pipeline uses STAR + RSEM for quantification of genes/transcripts.