Can I use a basic compound microscope to analyze the ploidy of plant chromosomes?
If you want mitotic chromosomes then you want to look at apical meristems — probably the primary root from germinating seedlings will be easiest.  Apical meristems are at the growing tips of roots (and shoots) and contain regions of rapidly dividing cells.  Since you can only see chromosomes as separate bodies when they are condensed for cell division this increases your chances of visualizing chromosomes.  

However, you will need to stain the roots to visualize the chromatin (material that makes up chromosomes).

This paper has a protocol for staining dandelion root tips, but I don't know whether you can obtain all the necessary compounds.  However, colchicine and para-dichlorobenzene probably aren't really necessary (they will increase the number of cells in mitosis) and are toxic so best to avoid those steps anyway. 

Note, it looks like you would also need a 100x oil immersion lens, which doesn't look like it comes with the microscope you listed.  

You may also find this paper useful as a starting point for making "homemade" chromosome stains. 

Good luck!