Give and take

To summarize the rule of magic:

That which which you give must be taken from elsewhere.

I'll be using a few examples for each kind of magic to explain. For each example a mage's area of influence would be limited to a range like that of the vitruvian man. I'll have you decide how quick or how tiring the whole process is.

Elemental:

Fire. In order to produce fire you have to take the heat from somewhere else and bring it together to perform what you require. Setting a log on fire would require you to take the heat from the environment or from living things, making things colder or freezing in them process if too much is taken. This could be done in reverse to cause the opposite effect, taking heat away, causing the intended object to freeze but making the air hotter or whatever you've chosen to dump your heat into.

Electricity. You don't need a storm for this. The earth's magnetism provides everything you need. Simply coil a bunch of copper wire around your mage's arms, have them swing that stuff around like madmen, and the magnetic field should interact with the coils in a sufficient enough level, however small, for the mage to collect the electromagnetism into their hands or something for use as an eventual shock. Alternatively, a staff coiled up in copper wire, or have a bunch of mages do this at the same time and have each of them hand over the electric potential to a single mage who will then be the one to fire off the lightning bolt.

Mages should be careful to not burn/freeze/electrocute themselves when taking/giving the various energies.

Healing:

Healing can be done in one of two ways. Using one's own flesh to heal someone, or using someone/something else's flesh to heal. You can also heal yourself by taking the biological matter from someone/something else. Healing magic requires medical knowledge to perform effectively, otherwise the healer may very well eventually kill someone due to blood clots forming or incompatible biomatter not agreeing with the person's immune system or veins or organs simply not working like they should. This is also why simply healing yourself or others on the battlefield tends not to be done, because you've no way to really confirm whether the flesh would be accepted or if you've done the healing correctly in the destination body in the heat of battle and as such it is only done in severe emergencies, where at camp you can use the flesh of relatives or those most likely to be compatible to perform the healing and have the time to confirm that the healing won't eventually kill them. You don't need the flesh or bone or organ to come from a still living person, you can simply use that which has been harvested beforehand though needless to say the matter still needs to be somewhat alive, which is where elemental magic and healing magic can work together in order to freeze or keep cold biomatter for later healing use.

Transformation:

Transformation magic throws the laws and practices of healing magic out the window, by intently taking in incompatible flesh from another species to give yourself or someone superhuman abilities, strength of a bear or the protection from the scales of a crocodile, or the night sight of the eyes of a cat. Transformations kill the animal the mage takes their newfound abilities from and eventually kills the mage or person that has taken on the bodyparts/form, and requires an exceptional healer to put everything back together in the right way which may still eventually kill the person who transformed due to the healer missing something.

Greater medical/veterinary knowledge will increase the effectives of both healing and transformation magic(and the survival rate of the patients/affected) while greater thermodynamic or otherwise scientific knowledge on the laws of the universe will increase the effectiveness of elemental magic.