Congrats on the new lathe. You made a good choice.
You have gotten an early lesson on how buying inexpensive tools can end up costing more in the long run. It's a recurring theme in any shop, and pretty much all of us have experienced it.
...Will definitely be using the expansion hold instead of the compression from now on...
Don't make any firm decisions at this point. Wait until you learn more about chucking. The expansion hold is more likely (in my experience) to have or cause problems than the compression hold. But the key to either type of chucking method is to make the tenon or recess match your chuck jaws. Equally important is knowing the difference between "tight enough" and "so tight it's breaking the wood". Personally, I prefer a tenon, but I've used a recess when it seemed appropriate.
For tool suggestions, I'll echo Carol. Doug Thompson makes great tools. They are mid-priced, but they are among the best you can buy. And there are plenty that are more costly that aren't as good, IMO. I'd also suggest not looking for a set, but instead buying chisels as you need them. If you want to turn bowls, a standard 5 or 6 piece chisel set quite likely will only have one or two tools that you'll actually use. Personally, my go-to tools are several bowl gouges, a couple of thick scrapers, a detail gouge, and (every once in a while) a parting tool. I've got skews that seldom get used. And a few cheap spindle gouges that I have no use for at all. That all said, a lot of us started out with the HSS tool set that Harbor Freight sells. I still use the parting tool from that set, and re-ground one of the (otherwise useless to me) scrapers into a dovetail scraper for shaping tenons to match my chucks. The rest of the tools got me started, but have since been abandoned or replaced with better tools.
And Ted is again spot on with his suggestions to invest in good sharpening gear. Once I discovered how a properly-sharpened tool operated, turning became a lot more enjoyable for me. The Oneway Wolverine jig is pretty much the industry standard. Great tool. I got a lot of use out of Penn State's knock-off version, which is serviceable, but not as solid as the Oneway jig. Other guys get good results with homemade versions. And although my current dry grinder has high-quality stone wheels, I plan to replace them at some point with CBN wheels. (I also use a Tormek wet grinder for my gouges, but that's a pretty expensive proposition for a new turner. I only have one because I found a deal on a used one that I couldn't pass up.)