see Are There Open Bioinformatics Projects That A Beginner Could Learn From? Project For A Beginner Bioinformatics Student Open Source Project For Beginner How To Get Started In Bioinformatics? Hi. A Beginner. Need Some Ideas For Project. ....
NGS Analysis might be a good place to start. ChIP-Seq analysis, RNA-Seq Analysis.
Go ahead and download some raw data (fastq files) from Encode or GEO. Look up a published paper and try to re-create the results.
Here's a good paper on ChIP-Seq analysis: http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1003326
You can try some Bioinformatics problems first: http://rosalind.info