Learn Hardcore Russian: Мне легко учить новые слова утром. - It is easy for me to learn new words in the morning.

Мне = “to me” (dative of я)
There is no nominative subject like я here. Instead, the whole infinitive phrase учить новые слова утром functions like the “thing” that is easy.

This is the same pattern you see in:

Мне холодно. – I am cold. (Literally: To me it’s cold.)
Ей трудно работать. – It’s hard for her to work.

Using Я would change the structure and meaning (see the next question).

Could I say Я легко учу новые слова утром instead? Is that the same?

It’s not exactly the same.

Мне легко учить новые слова утром.
- Neutral, common way to say: It is easy for me to learn new words in the morning.
- Focus: your feeling / level of difficulty.
Я легко учу новые слова утром.
- Means roughly: I easily learn new words in the morning.
- Focus: how you perform the action (you do it easily, with no effort).
- Grammatically OK, but stylistically it can sound a bit unusual; more natural would be:
  - Я легко запоминаю новые слова утром.

For talking about “easy / hard for me”, Russians strongly prefer Мне легко / трудно / сложно... with the dative.

What case is мне, and when is this pattern used?

Мне is dative singular of я.

Russian frequently uses Dative + a predicative word to talk about states, feelings, possibilities, and difficulties:

Мне легко / трудно / удобно / скучно / холодно / жарко / интересно...
Then often an infinitive follows:
- Мне трудно говорить по-русски. – It’s hard for me to speak Russian.
- Ему интересно читать по ночам. – It’s interesting for him to read at night.

So your sentence fits the pattern:

Мне (dative) + легко (predicative) + учить (infinitive)
→ It is easy for me to learn...

Where is the verb “to be” in Russian here? Why isn’t there a word for “is”?

In present-tense sentences like this, Russian usually drops the verb “to be” (быть):

English: It *is easy for me to learn new words in the morning.*
Russian: Мне легко учить новые слова утром.
(No separate word for “is”.)

This is normal:

Он врач. – He is a doctor.
Мне скучно. – I am bored.
Здесь тепло. – It is warm here.

If you put it into past or future, быть appears:

Мне было легко учить новые слова утром. – It was easy for me...
Мне будет легко учить новые слова утром. – It will be easy for me...

Why is учить in the infinitive? Could I say учу instead?

Учить is in the infinitive because after words like легко / трудно / нельзя / можно, Russian usually uses an infinitive to describe the action:

Мне легко учить... – It’s easy for me to learn...
Ему трудно вставать рано. – It’s hard for him to get up early.

If you say:

Мне легко учу новые слова утром. – This is wrong; you cannot mix мне легко with a conjugated verb like учу.

You can say:

Я легко учу новые слова утром. (see above),
but then you’re not using the мне легко + infinitive pattern anymore; it’s a different structure.

What’s the difference between учить, учиться, and изучать here? Why учить?

These verbs are easy to mix up:

учить кого? что? – to learn / memorize something specific, or to teach someone
- учить новые слова – to learn new words
- учить детей – to teach children
  In your sentence, you are learning specific words → учить новые слова is correct.
учиться где? как? чему? – to study (in general), to be a student, to learn how
- Я учусь в школе. – I study / go to school.
- Он учится русскому языку. – He is learning Russian.
  You cannot say учиться новые слова; that’s ungrammatical.
изучать что? – to study something in depth, systematically
- изучать русский язык – to study the Russian language (as a subject).
  For individual vocabulary items, учить новые слова or запоминать новые слова is more natural.

So for “learning new words”, учить новые слова is the right choice.

What case are новые слова in, and why that form?

Новые слова is accusative plural, used as the direct object of учить.

Breakdown:

слово – singular, neuter: “word”
слова – plural: “words”
новый (m.), новое (n.), новые (pl.) – “new”

For inanimate nouns, the accusative plural = nominative plural, so:

Nominative plural: новые слова
Accusative plural: новые слова

We use the accusative because we are learning what? → новые слова (direct object of учить).

Could I say учить новых слов instead of учить новые слова?

No, not in this meaning.

Учить новые слова – correct: to learn new words (you are memorizing specific items).
Учить новых слов – sounds wrong / unnatural here.

Reasons:

As “to learn / memorize”, учить normally takes its object in the accusative:
- учить правила – to learn the rules
- учить стихи – to learn poems by heart
- учить новые слова – to learn new words
Учить новых слов would sound more like you’re using учить in its “to teach” sense with an omitted object (e.g. “to teach (someone) new words”), and even then we usually keep both objects clearly:
- учить детей новым словам – to teach children new words.

So: stick with учить новые слова for “to learn new words”.

Why is it утром and not в утро, утро, or по утрам?

Russian has some fixed time expressions using a special form (historically instrumental) that functions adverbially:

утром – in the morning
днём – in the daytime
вечером – in the evening
ночью – at night
зимой, летом – in winter, in summer

So:

Мне легко учить новые слова утром. – It’s easy for me to learn new words in the morning.

Other options:

по утрам – in the mornings (regularly, on mornings in general)
- Мне легко учить новые слова по утрам. – I easily / it’s easy for me on mornings in general.
в утро is not used in this sense;
утро alone would just mean “morning” as a noun, not “in the morning”.

So утром is the standard, natural adverbial form here.

Can I change the word order? For example: Утром мне легко учить новые слова?

Yes, Russian word order is flexible, and all of these are possible:

Мне легко учить новые слова утром.
Neutral, very natural.
Утром мне легко учить новые слова.
Slight emphasis on утром (when it is easy).
Учить новые слова утром мне легко.
Emphasizes легко (it’s easy, not hard).

All of them keep the same basic meaning. Russian uses word order mainly for emphasis and information structure, not for grammar like English does.

Is легко an adverb or an adjective here? How does it work?

Formally, легко is the short neuter form of the adjective лёгкий (“light, easy”), but in modern Russian it behaves very much like a predicative adverb in sentences like this.

You mostly see it in constructions:

кому? + легко / трудно / сложно / удобно + infinitive
- Мне легко учить... – It’s easy for me to learn...
- Ему трудно работать. – It’s hard for him to work.
- Им удобно встречаться здесь. – It’s convenient for them to meet here.

So in practice, you can think of легко here as “easily / easy (to)”, a special word used in this dative + infinitive structure.