1.The doctor will speak to__came first(whoever, whomever) 2.She can't remember___it was(who, whom)?

There are formal rules for who/whom, but if you're American like me, you may hear people using "who" for both cases. If English is not your first language, this won't help, but if you are American, this tip may help:

Change the sentence into a question, and try to answer with "he" or "him." Both hiM and whoM end in M, so if "him" fits, use "whom."

Examples:

The doctor will speak to ___ came first.

Which person will the doctor speak to?

He. (sounds wrong to me)

HiM. (sounds right, so whoM -> whomever)

She can't remember ___ it was.

The forgotten person is he.

The forgotten person is him. (hiM -> whoM)

___ washed my car?

He washed my car. (sounds right. Use "who.")

Him washed my car.

For the third question, @Politically's answer is perfect. Except that Americans do put sugar into what they call cake batter.

The grammar rule is that it is 'who' when it is the subject and 'whom' when it is the object.

So in the first the blank is the object place

In the second the blank is in the subject place

Your third uses a different rule. The verb has to match the subject. 'Go' matches a plural subject(they go) and 'goes' matches a single subject (it goes). 'Either' means milk or sugar but not both so it is always single.

By the way, tell your teacher to use better examples. We never put sugar in batter.