Lee Loader

As I become comfortable with my ability and familiar with the Pro 4000 loading 9mm, my next step would be .357/38.

Besides the die set and the shell plate, would I need to acquire any other press parts?

I haven't used my Pro 4000 in a year, or so, so I had to look at it, but you're correct, you will just need the die set, and shell plate, for 9mm to .38/357.
You might want to consider this.
Lee just came out with some new Powder Thru Expanding plugs, for the Powder thru Expanding Dies. I believe they have them for quite a few calibers, just not all, yet. All of them except the one for 9mm are like the Lyman M die, which makes a step at the end of the case, instead of just a flare. It really helps the bullet, especially flat based, coated, or cast bullets, seat into the case straight. The 9mm plug is very similar to the existing plug, but pulls less when removing it from the case. They are worth money, around $4.00ea, plus shipping.
They did not change the part numbers, so you need to email, or call Lee and they will make sure you get the new parts, not the old ones.
The 9mm plug is part#- SE2413, and the .38Spcl.357Mag is SE2408.

Thank you!

You're welcome!

I'm thinking that for .45 (ACP or Colt depending on which I start with), it'd be the same thing (die set & shell plate), but I'd switch over the large primer arm (included with 4000 kit).

Yes, just make sure you pay attention to the .45 cases, as some brands are Small primers.