How can I upgrade to a better gear range?
No, your derailleur won't work with 9 speed casette. Well, it could work on 8 sprockets of the 9, but that won't help you.

You certainly can buy a new 8 speed casette from the Claris line (or MTB) but you cannot take just any, you cannot go over your derailleur capacity and over the largest sprocket size supported so you are out of luck for the modern 32T casettes. Unless you find a good deal on a more recent 8-speed RD (like the new Claris 11T-32T, however, it is also sold as  11-28T, 12-25T, 13-26. The 28T is one tooth over your RD but might still work.). 

Changing one chainring should work, check it is a compatible one. That would be cost-effective. If you have the Sora 3300 FD, it has the capacity of 15T, so with 52 big ring you can go to a 37 small ring.

Otherwise, you could always buy a whole new crankset, for example, a compact one (50/34), those are the most common and certainly would bring you a better gear range. But I am not sure about the 16T difference with your front derailleur. There are two versions of RD3300, one is SS and one is S
GS. One supports the front chainrings difference of 14T, the other 22T. If you have the GS (total capacity 37T), you are good to go, but with SS (total capacity of 29T) the shifting might have some problems.

Again, better to take an 8-speed one from the Claris range, or an older 8-speed Sora. But, buying a new crankset will be the least cost-effective. Unless you find a good deal on something older (but better unused).

This kind of crankset would be a good candidate https://sprockets.uk.com/shimano-rs200-8-speed-compact-chainset-black-170mm-34-50t/ (I mean RS200, this specific 170 mm might be too short for you). But for the SS version the difference is officially too big.