How could I minimize the aircraft total takeoff fuel consumption, taking into account all fuel used from start of take-off roll until reaching a safe altitude?
For example, if I fly with flap retracted and the runway is long enough to takeoff, does that mean I could use less energy than flying with flap extended, since I have less drag?
It's not specified in the question, so I'll assume that it's a gasoline-powered fixed-wing SEL airplane (aircraft is anything which flies, including balloons, rotarywing, etc...) on a standard climbout from a standard (but looooong) runway, etc... I'm also assuming you can't do exotic things like change the engine compression ratio.
I will also assume that efficiency in this context is only energy consumed per unit height. If we want to define efficiency as energy consumed per unit distance traveled, then the problem moves in a different direction and becomes much more complicated.
In my responses below, please understand that these are first-order effects, but second-order effects can lead to modifications. For instance, maximal efficiency might come at a slightly faster/slower airspeed in order to allow for a slightly faster/slower engine RPM.
Also, in general, fuel is not the important consideration for takeoff. Safety is. It's better to burn a few mL more of fuel and keep the airplane from disaster (which would presumably result in wasting a lot more energy).
The mixture is the biggest control on efficiency. In order to achieve maximal efficiency, you need to run as lean as possible. Preferably lean-of-peak, if this can be done. We're talking 20% savings or more here.
If you're not taking off at high altitudes, e.g. in the mountains, don't ever try this if you value your airplane engine's life, and as a consequence your own. The engine needs the enhanced cooling from the rich fuel/air mixture. Only at high density altitudes can/should the engine be run at full power at extremely lean settings.
The throttle should be fully open, which will provide the least restrictions to the engine breathing.
If you have a constant-speed prop, then it should ultimately be run as slowly as possible. This gives maximal time for the burning fuel energy to be extracted, as well as minimizing the number of engine cycles and friction.
The RPM should be advanced at such a speed as to provide peak propeller efficiency for the advance ratio.
Use the lowest octane gas you can, as this has more energy content than higher octane. Please note that lower octane increases the propensity of the engine to knock/detonate.
Fly the plane at Vy, which is best climb airspeed. This is the airframe's most efficient operating point.
In general, don't use them for optimal efficiency. They increase drag more than the increase lift. That's why we like them on landing because we can come in at a steeper angle of attack.
An argument could be made for rapidly "pumping" the flaps in order to pop the plane off the ground, thus eliminating rolling resistance at an earlier point in the takeoff roll. This technique was employed to great effect at a recent STOL competition.
I think you’re asking about how to get to a safe height with minimum expenditure of fuel. As you suggest, the aircraft will generally be more efficient in flight with the flaps retracted, and certainly before liftoff there will be significantly less drag than with full flap. The downsides are that the takeoff run will be longer, and the takeoff speed will necessarily be higher. If what you mean to ask includes reaching normal flight conditions, flaps up, cruising height and nominal airspeed, then a flapless takeoff is very likely to use less fuel. On the other hand, if you consider 10ft to be a safe height and you have an aircraft with small wings and huge flaps then the opposite may be the case.
You can do two things:
I used to go a little farther than that when I was flying regularly from an uncontrolled airport with an 11000 ft runway. I would take off using climb power (24" MP), more or less using "flex thrust" like a jet. I would use 2000 ft of the 11000 ft runway instead of 1000 ft. I probably save a thimblefull of fuel or so, who knows.
Configure airports and planes in such a way that take-off catapults, similar to those used on carriers, can be used to accelerate them up to take-off speeds while using less onboard fuel than would otherwise be used.
Note that the greater distance available with such a ground based "catapult" means that the device can act over both a greater distance and longer period of time, which would reduce the strain on the equipment and passengers.
The complexity of this system would be cost prohibitive to set up and maintain, but if you only care about reducing the fuel usage during takeoff, this would do so.
Caveat: I've been reading a lot of worldbuilding posts recently, and this post was likely highly influenced by that.
Use lighter than air aircraft (aerostat) filled with helium or hydrogen. They have other shortcomings but should not use much energy just for lifting up.
