Yes, you should get a new set of dies meant for a progressive press. This will mean that you don't need an expander die unless a lead bullet specialty like the Lyman M-die. However, you will need the Hornady expander plug for inside the powder drop and the stop bracket for adjusting belling. You may also need a pistol caliber powder measure insert that does not come with the press but is needed for charges around 5 grains or less.
I have taken to buying dies individually to get best of breed in each one and to eliminate dies I won't use. RCBS is a mainstay for me but I am impressed with Lyman as well. I have various Reddings, which are nice, but I don't think that expense is always necessary. Others may be suitable, but these are my own favorites that I am comfortable recommending. Lee is not a good choice since they don't have threads long enough to fit an LnL AP very well.
I am not a fan of Hornady dies except that, if I use a combo seat/crimp, the Hornady works well enough and has a good adjustment design. For lead the RCBS Cowboy die cannot be beaten. The Hornady is just easier to find in stock as a separate die. I don't always use a combo die and might have a Redding Profile Crimp Die added. I have a bullet feeder station in a couple calibers, which forces me to use a combo seat/crimp. I also use an M-die which forces out any powder check, so in my loads I look first for powder charges with good volume that I can readily see at the bullet feeder station, up front at #4. The manual specifies station numbers, and it will be good to both refer to them in that sequence and to mark or label a couple. I have stickers on 3 and 5 and can extrapolate the rest.
You can avoid some duplication in these dies by buying a couple sets of Redding spacer rings that boost the dies up to magnum level, e.g. using 38 Special dies for 357 using exactly the same settings plus the increment of the spacer. Doing this pretty much requires that you have set screws on your lock rings You may need a few extra, and Hornady's design with tangential screws is a good one that won't damage die threads. There are workarounds for other lock nuts, but you will do well to use the Hornadys on a Hornady press.