Tools to analyze RNA-seq data
You will likely need a tool to "map" the reads on the reference genome.
You may find such a reference genome, together with annotations, here:
ftp://ussd-ftp.illumina.com/.

Mapping tools such as bowtie2 or bwa take fastq files and reference genomes and output mapping results in a format called sam.

You then have a lot of options to estimate gene expression.


You can write your own algorithm to parse sam format and estimate normalized read counts on each gene.
You can combine more or less low-level tools such as samtools, pysam, htseq with some scripting to do this.
You can use tools that do the counting (like bedtools ot htseq-count) and differential expression analysis (like deseq2).


In the last case, I would advice to start from the documentation of the final tool to find out what are the tools you need to generate the output of the preceding step.

It is very likely you will use some R or Python, or use the web platform galaxy for some of the steps.

Edits

As mentioned by @scribaniwannabe in this answer, the paper about the Tuxedo suite of tools gives a good example of the steps to carry out an RNA-seq analysis using recent tools (as of October 2016).

As @Student T reminds in this answer, RNA-seq data contain reads that can come from exon-exon junctions, so the read mapper has to be set up in such a way as not to discard reads not mapping continuously on all their length on the genome. To my knowledge, HISAT2 and CRAC do this by default. Bowtie2 needs special settings.