The press in the Deluxe kit is the standard turret press, it's a light duty press, but it's a little cheaper if you're trying to keep your start up cost down and since you're reloading just pistol rounds it should provide adequate service. One of the reason I chose the Classic Turret was I knew I would be reloading rifle rounds too. Plus I prefer cast iron and steel over the aluminum.
If you're getting Lee die sets you wont need the shell holders for the press, one is included with each set of dies. You will need the shell holder set for the hand primer though, Lee part #90198. You'll need both the 9mm and 380 die sets as they are a little different in size. Looking at the specs in my Hornady manual the 9mm case is .380" at the neck and tapers to .391' just above the rim. The 380 is .373" and is straight wall to just above the rim. If you want to try to reload both with the 9mm dies, I would suggest making some 380 dummy rounds and see if they chamber in your pistol. I really recommend getting both die sets though, it will save a lot of set-up time when switching calibers, you will have to re-adjust everything when switching.
There's a few thing you'll need that doesn't come with the kit. The hand primer and hand primer shell holders. A set of calipers to measure cases and loaded rounds, I prefer dial, but a lot of people use digital because they're easier to read when starting out. A bullet puller for the mistakes that are made now and them. I have two, the Hornady Cam-Lock and a kinetic bullet puller from Cabela's. Both do the job the kinetic is just cheaper. The lee Case length gauge for each caliber, about $4-5 each.
The Lee scales that came with my kit never worked right and I've read a lot of complaints on them too. So I replaced them with the RCBS 505. I really don't recommend the digital scales, the good digital are high priced and the cheap scales are cheaply made, I would recommend the RCBS scales instead. You should have a good set of balance beam scales to periodically check the digital scale for accuracy anyway.
Ive been using the $14 Iosso case cleaner kit for my brass, most prefer the tumbler, media and the hours involved with it. The liquid cleaner takes seconds to clean the brass and is cheaper to start out.
I'm sure there's a few other items needed, but can't really think of them at the moment.
Iosso liquid case cleaner
Kinetic bullet puller
RCBS 505 scale
. A little high, but well worth the extra cost.