Lee hand press kit
Lee dies (include shellholder and a dipper)
powder (you'll need different kinds for .45 and .223, and if you're going to use the Lee dipper, choose one that has a listed load for using the dipper)
primers (small rifle for .223, large pistol for .45 ACP)
bullets (generally speaking, lighter bullets for the caliber will be less likely to blow up your gun if you make a mistake; but better not to make mistakes!)
For a beginner, some kind of introductory manual is essential. I learned from the Lee "Modern Reloading" 2nd edition, and it is very good. There are other good manuals as well, of course.
This is the absolute minimum, but I think that would be enough to start making ammo.
I would highly recommend:
-universal reloading block like the MTM or Hornady models (about $5-8 each, one is enough to start with)
-calipers (I have some all-stainless vernier style calipers that I got from Lee Valley Tools [unrelated to Lee Precision] for about $20. They work great for me, but you must have good eyesight to use them. Most people would get a dial or digital type caliper; I didn't because quality models cost a lot, and cheap models break quickly
-0000 steel wool for cleaning cases
-case length gauge for each caliber, case trimmer, and chamfer tool - almost essential for .223 and other rifle cases, not very important for .45 ACP and other low-pressure pistol cases
Extremely helpful, but can buy after the above:
-powder scale
-powder measure
-bench mounted press
FYI, when people read "Lee Loader" they think of a different device - you're talking about the hand press, which is pretty much usable like any other press. I started with a Lee hand press and it was good to start with. Now that I have a bench-mounted press I use that for all standard tasks, but the handpress is still used for depriming sometimes (universal decapping die) and could be used for priming if I wanted to (with the ram-prime that comes in the hand press kit).
If money and space aren't super tight, I would recommend you instead get the Lee Challenger kit, or a similar kit from any major manufacturer. The hand press gets very tiring for doing more than 20 or so cartridges.