Good, VE2RF, sounds like you're on your way. Looking back over the last 40 years, I know what powders I should have started with.
With the powder measure that comes with a Loadmaster, I do believe I would use HP-38 and/or W231 for the pistol cartridges. Both powders are the same and both are distributed by Hodgdon, but W231 has a Winchester label on it. Identical powders other than the label. Not only is this powder a common standard bearer for pistol, but because it is a spherical ball powder it meters extremely well in Lee equipment.
For your rifle calibers I believe I would stick with another ball powder for ease of measuring. Winchester 748 is the Winchester factory standard for both your cartridges, 223 and 308. There are dozens of other choices that also perform extremely well, but I would load a few thousand rounds with a Lee-user-friendly powder before I start experimenting with other choices.
Fine grained ball powders can sometimes drop a granule or two from the Lee powder measure. Just check to make sure everything is tight, and gently blow the droppings away from the equipment so they don't upset the system. It's totally surprising how a tiny little granule will throw a case or a primer out of whack. But blow it out of the way and you're good to go.
For primers, use standard primers from Winchester, CCI, or Remington. You can't lose with any of them. Make sure you seat the primers fully in the pocket for 223 if you're using it in a semi-auto. If you seat the primers fully, you won't have to bother reading the online horror tales and urban myths about slam fires.
Read up about crimped primer pockets. If you wish, you may consider starting out with non-crimped commercial brass until you learn more about your craft. Believe me, there is plenty to learn and the Loadmaster does require you to focus. After you get the feel of primers being properly seated in commercial pockets, you can use your leisure time to cut crimped pockets. It takes a good sense of feel to tell if you have reemed or swaged the pocket properly.