When rough milling boards from a green log, is there a good rule of thumb or something to determine how much thicker I should slice them than my desired target thickness? Does it differ depending on the cut? I want to account for warping during drying but also minimize planing waste.
If you've got a lot to spare, then overdo it. Curves, twists, warps etc will always kill your yield. Flatsawn is apt to be less stable (esp with respect to cupping) than quartersawn.
The milling marks you'll get from the sawmill will have an impact, too. Really rough, and you'll want a bit more thickness in the rough size.
Regarding the final use... short pieces? long? gotta be exactly 3/4"? If you're flexible and don't need long pieces, you might go with a little less thickness.
Most good rough lumber is 1/4" over the intended finish thickness. (That's dry, so it started a bit thicker.)
So, no rule of thumb for you, except to maybe overdo it your first few times and see how it goes.
I believe there is just no easy "rule of thumb" when deciding what thicknesses to cut green lumber in an attempt to obtain a certain machined dry lumber thickness, without a lot of waste. Many factors can be considered for the best results.
If a log is in a lot of tension, it will be moving while you're trying to cut it.
When your desire is a specific finished dimension with minimal waste as you mentioned, you don't want to cut too thick for fear of waste or too thin making it unusable for its purpose.
No one board, cut green, is exactly 1.125" thick its whole length. If that cant is moving around a lot as you try to cut it, due to relieving internal tension, you may need to start out at 1.375" to get that 1.125" at the other end. Length is important. Short logs are very forgiving as Aloysius pointed out above. Long boards of continuous thickness require exceptional logs.
Every log is different. If it grew on the side of a hill competing for sun light, you may have to cut at 1.50" to get the 1.125" that you're after.
Mill blades are effected by dirt that may be hidden in the bark. A hardened knot can alter the track of a blade.
A dull blade can dip or rise causing more than just a machining mark defect to plane out of a board.
Operator error is a big problems also. A chunk of bark underneath one end of a cant will get you a board thick on one end and thin on the other. Very wasteful in wood and power to plane it out to a uniform thickness.
A cant is moving around as you release tensions with every board you cut, causing errors in thickness. This is especially true when your cant is thin with no mass and you're trying to split it into 2 equal thicknesses. (Splitting hairs) You are likely to end up with two unusable thicknesses for your purpose. Leave those thick for mantles or something.
I agree with Aloysius, err on the side of extra thickness, but not just for cup and grain abnormalities, also operator error rears its ugly head a lot. Especially with band-saw mills.