So you have taken the bass for a setup?
Anyhow, upgrading won't make you sound more like your favourite bass player. Even getting the same bass as he/she plays won't do that. Most of the tone is in the fingers. For example, it took me five years and thousands of hours to get what I call a decent sound out of a bass guitar, but then again, I'm very picky when it comes to that. And after fifteen years, I still do about 30% of my playing with a squier, but I admit it has been modded a bit.
What you have is a decent instrument all by itself. If you haven't already taken it for a setup, you could ask the tech for tips. New nut and new bridge won't have a huge impact on the sound, but the new nut can make your bass play better. Some people could say they make no difference to the tone. Fret leveling is also sometimes a good idea and some people like the pots replaced along with the pups. But remember, the tech is trying to make money too, so he might suggest you get every single part changed and you end up spending a lot money for improvements you might not even notice while playing with a band.
But I'd say buying a more powerful amp would not be a bad decision at all. Good amplifier will make a bass sound a whole lot better but most of all, it'll allow you to do the only thing you really should be doing: playing with another musicians.
I don't think there is much point in buying an expensive bass in the first few years because it's actually really hard to even know what kind of instrument you'll end up liking in five years. After four years of playing I bought a warwick which I thought to be the coolest thing ever. Few years later I discovered that I hated it.
Nothing else than practice can make you sound good. But then again, if you have money to spare and wan't to buy an expensive bass then there is no reason why you shouldn't. But don't expect it to be a free ticket to sounding good because at the end of the day, it's just another hunk of wood with some metal wires on top.