so... as stated above make sure your gun is rated for 5.56. the case dimensions are the same as 223, however 5.56 is loaded to higher pressures and is not safe in a 223 rifle.
So to answer your actual question, its complicated.... it depends on several things, most importantly what platform you will be shooting, semi or non semi. After that it comes down to budget and personal preference. I cant give you a direct answer without knowing but I will try to explain as best i know how. It comes down to these following questions
1 semi auto or non semi auto
2 do you want to crimp
3 do you want to neck size (no an option for semi auto)
neck sizing does not work well in semi auto guns, but can give more accurate round in a bolt gun with fire formed brass from your gun. if you want to neck size, get the lee 4 die ultimate set. their neck sizer is excellent and great for beginners to neck sizing.
crimp. there are 3 styles of crimp that I know of, roll, taper, and the lee factory crimp. my understanding is that with roll and taper you need bullets with a cannalure and you have to crimp on the cannalure to use the crimp. the lee factory crimp die you can crimp what you want where you want but it is an extra step and not built into the seating die. You don' have to crimp, but I like to.
my personal recommendations
1 non semi auto, if you want to neck size, hands down, get the lee 4 die ultimate set
2, non semi auto, don't care about neck sizing/crimp, hornady 2 die set
3 non semi auto, don't care about neck, want crimp, hornady 2 die set buy lee crimp die
4 semi auto dont care about crimp, hornady 2 die set
5 semi auto, care about crimp hornady 2 die set, buy lee crimp die
I have the lee ultimate 4 die set for 223. It has been a good die set but i regret it because shoot semi auto and i never used the neck sizing die, it was a waste. If i had to do it again i would get the hornady 2 die set and buy the lee crimp die. I like the hornady set because of the bullet allignment sleeve, and that you can buy more seating stems for different bullet styles and a micrometer adjuster add on the the bullet seating die. the lee crimp die is the best crimp IMHO, and its only like $14.
Lastly, I like the hornady dies for the reasons listed above, no brand preference. With that said, in all honesty any 2 die set will get the job done. It just depends on your budget, and what features you want/how you want to load.
sorry i know it was a bit long winded, hope this helps clear things up, not make it more complicated