Most public datasets are just fastq files, so your life will be easier if you can get access to either a cluster or at least a single larger server (e.g., 32 cores and a 100 gigs of RAM).

If you're doing a PhD then you should have some affiliation with a university or research institute (after all, a company can't award a PhD) and they should be able to help you find compute resources locally.

• to mine data from literature transcriptional analysis (microarray/RNA-seq)
• I might also have to analyses a couple RNA-seq experiment myself.

Those are two very different requirements. For the first you don't need a lot, normal laptop should be fine. For the second it also depends on how many samples (and how many reads) you want to process. You can also have a look at Galaxy to work on.

If you need to analyze RNA-seq, would you exactly repeat the pipeline from the original paper or can you do something else? There are "more recent" approaches which are pretty fast and don't need a beast, have a look at Kallisto/Sailfish/Salmon for pseudoalignment and read counting. Those might save you a headache.

I'm guessing based on this:

but I have read that for RNAseq, especially on human data

Your new role will require you to analyse human data?

If that isn't the case and you're based in the UK, there are facilities like MRC CLIMB which provides free virtual machines for microbial bioinformatics (it's based in my lab at Uni) http://www.climb.ac.uk/

Similarly, if you have absolutely no access to any physical hardware via the uni or otherwise, you can consider Amazon S3/AWS cloud computing solutions. If you aren't going to need the hardware indefinitely, some on-demand VMs might be a good compromise - though you'd have to do the maths to figure out whether its more cost effective.