The Warframe Wiki has a very nice explanation of what is needed to level mods.
First the economical answer, the cost of fusing mods in credits are directly tied to the amount of benefit it can potential supply. So using the most optimal fusion of mods from a mod perspective will automaticly also be the best solution on a financial view.
To level a mod you need to add a certain amount of energy to it, this energy comes from fusing other mods with the mod you wish to level.
There are 3 factors when fusioning mods:
- Level
- Rarity
- Polarity/Type
Level
Energy required to level a mod is doubled each level and they start at level 0, except for Fusion Cores, and goes up to 10.
Energy they can supply for fusion is base + ½ * level * base.
So if at level 0 it supplies 2 energy, then at level 1 it will supply 3 energy, at level 2 it will supply 4 energy etc.
The level is the number next to the rarity, no number is equal to level 0, or the number of white stripes on the left side.
Rarity
There are 3 types of rarity:
common has a multiplier of 1, uncommon 2 and rare 3.
Calculating Energy needed to level
With these 2 factors we can calculate the amount of energy needed to get a mod to the next level, to make the math easier later when we get to Polarity/Type we make sure it is a multiplier of 4.
E = 4 * R * 2^(L - 1)
- E is energy needed to level from precious level.
- R is the rarity (Common = 1, Uncommon = 2, Rare = 3).
- L is the level.
This gives us the following chart:
Level
1 2 3 4 5 6 7 8 9 10
Common 4 8 16 32 64 128 256 512 1024 2048
Uncommon 8 16 32 64 128 256 512 1024 2048 4096
Rare 12 24 48 96 192 384 768 1536 3072 6144
Calculating Energy supplied by mod
We can also calculate how much Energy a mod supply at different levels:
E = 2 * R(2 + L)
- E is energy needed to level from precious level.
- R is the rarity (Common = 1, Uncommon = 2, Rare = 3).
- L is the level.
This gives us the following chart:
Level
0 1 2 3 4 5 6 7 8 9 10
Common 4 6 8 10 12 14 16 18 20 22 24
Uncommon 8 12 16 20 24 28 32 36 40 44 48
Rare 12 18 24 30 36 42 48 54 60 66 72
With these 2 charts you should be able to calculate how many mods are needed to upgrade a mod, but there is the 3rd factor.
Polarity/Type
Polarity/Type Is divided into 3 groups when fusing:
- Same mod (duplicates)
- Mod with same polarity or Fusion Core
- Mod with different polarity
Two of the same mods give 100% of its energy.
So fusing 2 level 0 common mods of the same kind it will transfer 4 energy from one mod to the other, resulting in it gaining g enough energy to reach level 1.
If you then want to get it to level 2 you will require 8 more energy which is equal to 2 more of the same mod
Mods that have same polarity or Fusion cores give 50% of their energy.
Looking at the chart we can see that a level 5 uncommon Fusion Core supplies 28 energy.
It will get a 50% penalty to energy supplied to it will add 14 energy to any core.
So when fusing it with a level 0 common core it will gain 2 levels (4+8 energy needed) and have 2/16 of the next level done.
Mods with different polarity gives 25% of their energy.
Having a rare level 0 mod that will provide 12 energy and fusing it to a common level 0 with different polarity will give it 3 energy, so it will be 75% of the way to level 1.
So the most efficient way of leveling mods is to use duplicates, once you get spare mods (don't need any more/higher level of a mod) you can use same polarity and fusion cores to level other mods.
My personal recommendation is not to use use spare mods/fusion cores to level common mods. Most of them drop relative often and you will be better suited waiting for some duplicates to drop and using spare mods/fusion cores on rare mods where you won't be seeing as many duplicates.
Best Answer
There isn't one. Elemental damage is basically a more complicated version of rock-paper-scissors. This means that the strongest elements against a particular type of enemy will be weak against others.
There are primary elements and secondary elements, which are combinations of two primaries. The secondary elements have more pronounced rock-paper-scissors effects.
There are also physical damage types, which don't combine.
Each element and physical damage has an effect (often called a proc) - for instance Heat can set the target on fire, Cold can slow them down, Impact can knock them over. The status chance of a weapon is the probability this effect will happen. For some enemies the effect is more useful than the base damage.
Mods applied to a weapon are combined left to right and top down, with the weapon's inherent damage type applied last. The first two primaries will combine, a third primary will apply after that and a fourth primary will combine with the third to create two secondary effects.
However, lots of different elements can weaken a weapon, as they split the status chance. A 25% status chance weapon with 2 elemental damage types will roll for the ¼ change of having an effect, and then roll again for which of the elements to apply (with the relative damage as the probability).
Physical effects are in this pool too, so pure elemental weapons with high status chances can be extremely effective with the right mods as they don't dilute the effect pool.
Some weapons have guaranteed status effects that always applies. For instance the Lenz always has a Cold slow down effect, regardless of how you mod it.
You want your weapons to be complementary, rather than stacked - if your primary mows shields you want your secondary or melee to strip armour.
The most consistent all-rounder is probably Viral (Cold + Toxin), but it takes a big hit against Infested (a Toxic Ancient can immunise all of them) and some Grineer. The hardest to use consistently is probably Radiation (Heat + Electric), but it's also the best against the highly armoured Grineer variants.
Corpus use shields, so bring Magnetic (Cold + Electric) to strip shields or Toxin as it bypasses shields with a damage over time. They're resistant to Corrosive. They're also resistant to Gas, but it has a secondary Toxin effect that bypasses shields - if you have a very high status change Gas becomes very powerful. Impact does the most physical damage to their shields.
Grineer use two kinds of armour, bring Corrosive (Toxin + Electric) for most and Radiation (Heat + Electric) for the Elites/bosses. They're resistant to Gas and Magnetic. Puncture does the most physical damage to their armour.
Infested are fleshy, so most are very vulnerable to Heat or Gas (Heat + Toxin). Higher level ones (Ancient/Mutalist) aren't though, you'll need Corrosive for them. They're resistant to Toxin and Viral. Slash does the most physical damage to their flesh.
Corrupted enemies are a mix of all of the above, bring a mix of weapons and abilities.
If you can rely on the effects happening (high status chance) these change slightly, for instance Slash becomes more universally powerful because its bleed over time effect will hurt everything.
As a general rule: