start by determining what bullet you want to use. either it has shot pretty well for you in the past, or you like some ballistic properties of it, or you are looking for killing reputation, whatever. so, now you have your favored bullet. if you are just starting out, go ahead and buy a second choice bullet to have on hand in case you find something w/ your first choice bullet that doesn't work. preferably in a different weight.
so, now that you have the bullet, crack a speer #13. i have always had best luck w/ the top powders they list for that bullet. so grab a pound each of the top 2 listed powders for your chambering and bullet weight. cross check that selection to make sure they will work w/ your back up bullet - they probably will, but as reloaders go, you will likely find another powder that interests you. optional purchase, but if you really get into handloading, you'll eventually have it anyway.
for primers... a brick of cci 200's for the 308 - unless one of your powder choices is a ball powder (like win 748), in which case you will also add some cci 250's. go ahead and grab a brick of cci 400's (add 450's if you'll be loading a lot ball powder) for your 223.
and you're all set.
-or-
you can bypass lots of this by asking for favored start components... for the 308 you'll get a lot of 'varget' and '165/168' responses - this helps you quickly narrow down your choices. for the 223, you'll get lebenty-eight billion different suggestions. pick one that makes sense to you and run w/ it.