Hey there NyuBrew and welcome.
We all have to start somewhere. Not having used your Breville Im no authority but I understand its a fairly solid machine, although built to "consumer" specifications, including an "advanced crema system" (pressurised portafilter). A big improvement may well be modifying the portafilter back to "unpressurised" -see numerous CS threads on this topic.
I think all CSers would recommend steering clear of pre-ground coffee as it would be stale before you use it. A reasonable quality grinder and your plunger would make better coffee. Therefore, if you want to make a nice brew, freshly ground beans (beans roasted by an artisan roaster within the past 3 weeks and ground on demand seconds before brewing) is pretty much essential. As has been said here before, your grinder is as, if not more, important as your machine.
Only you know how much you are able to spend or if you are able to save up in a reasonable period for a decent grinder. But generally a price tag from around $300 would be expected for good "starting point" grinders (eg Iberital Challenge, Ascaso I-mini). Having said that, many CSers have used Sunbeams EM0450 / 480 - priced from less than $150 - which are apparently capable of giving acceptable results for espresso, albeit with quality control problems and a relatively limited lifespan. Alternatively, if you are able to save up $700 +, you would be able to get a high quality grinder that would last the long term.
I suppose the better the grinder you get at this point, the better it will match up with a good quality machine when you eventually upgrade. Being a student, however, it may be that you have to make compromises at this stage and use your money for more essential things (yeah, I know, whats more essential than quality espresso ;D).
Why not have a chat to a sponsor or two (see left of page), they may give you good advice / options.