I've been ~5x in the last 3 years, and have always had a great time! By chance, I had also just put this together for a Chicago friend who went recently, and while jealous stuck here in Chicago, hope you have fun!

-Wasn't sure where you were staying, but if you're able to connect with the Budapest Party Hostel folks, they are awesome people who will show you a fun time, although I'm also convinced Aussies are born with 2 livers!
-Definitely walk up the hill leading to the castle. There is a funicalur trolley as well, but I actually saw a lot of cool stuff walking, and there is a really nice church there too.
-The St. Steven/Istvan Basilica is impressive! It is a long climb to the balcony area, but well worth it as it has some of the best views of the city.
-The Danube/Duna river is also amazing, and highly recommend the boat cruises as a great way to see the city, especially in summer time. My parents did one of those fancy food ones; I did the NYE party one with ~30 friends, and both are really fun
-Geller Spa is crowded, but also very worthwhile, and if possible it is best to pre-purchase tickets so you can walk in
-Parliament area is really nice, as are the bridges.
-Bors Restaurant is some of the best sandwiches I have ever had! Michelin-starred chefs who wanted to make good food affordable, and a soup + sandwich will cost all of ~$7
-Ditto Zing Burger
-There are a surprisingly good amount of food options. I am not a huge goulash fan personally, but we ended up having Thai, Caribbean, Vietnamese (pho hangover cure ftw), Italian, and burgers, all of which are very affordable
-The ruins bars are really fun! I am going to mangle the spelling, but Fogaz and Szimplot are two of the better ones
-There is also a really nice outdoor food market - not the Grand Market Hall - that I am drawing a blank on the name, but any local will know exactly where it is

Edit: Sprack had some awesome recommendations too, and fully agreed! As an aside, they have some bizarre drink in the palinka/rakia vein similar to Malort, which didn't bother me like a tequila shot, but was perplexing.