Η μετακίνηση προς την Αθήνα είναι πιο εύκολη με το τρένο.

Η is the feminine singular definite article, meaning the.

It appears because μετακίνηση is:

• a feminine noun
• singular
• the subject of the sentence, so it is in the nominative case

So:

• η μετακίνηση = the movement / the travel / the commute

Μετακίνηση literally means movement or relocation, but in everyday Greek it can also mean:

• travel
• getting somewhere
• transportation
• commuting / moving from one place to another

In this sentence, it means something like traveling to Athens or getting to Athens.

It is a fairly general noun, not just walking or physically moving your body.

Because προς takes the accusative case.

The noun Αθήνα is feminine singular, so:

• nominative: η Αθήνα
• accusative: την Αθήνα

So after προς, you need:

• προς την Αθήνα = toward Athens / to Athens

This is a very common pattern in Greek:

• προς + accusative

Here προς means toward or to, showing direction.

So:

• προς την Αθήνα = toward Athens / to Athens

Compared with other options:

• σε / στην Αθήνα
  usually means to Athens or in Athens, depending on context; very common in everyday speech

• για την Αθήνα
  often means for Athens, bound for Athens, or heading to Athens

• προς την Αθήνα
  can sound a little more directional or slightly more formal/written

So this sentence is natural, but προς is a bit more formal than στην in many everyday situations.

Because εύκολη must agree with μετακίνηση.

Since μετακίνηση is:

• feminine
• singular

the adjective must also be:

• feminine singular

So:

• masculine: εύκολος
• feminine: εύκολη
• neuter: εύκολο

That is why the sentence has:

• η μετακίνηση είναι πιο εύκολη

Yes, both are possible.

Greek has two common ways to form the comparative:

• πιο + adjective
  → πιο εύκολη = easier
• comparative form in -ότερος / -ότερη / -ότερο
  → ευκολότερη = easier

So both of these are correct:

• Η μετακίνηση προς την Αθήνα είναι πιο εύκολη με το τρένο.
• Η μετακίνηση προς την Αθήνα είναι ευκολότερη με το τρένο.

In modern Greek, πιο + adjective is extremely common and often feels more natural in everyday speech.

Είναι is the 3rd person singular of είμαι = to be.

Here it links the subject to the description:

• Η μετακίνηση ... είναι πιο εύκολη
  = Traveling ... is easier

So είναι works just like is in English.

με means with or by, and here it expresses the means of transport.

So:

• με το τρένο = by train

Greek often uses με + article + noun for transport:

• με το τρένο = by train
• με το λεωφορείο = by bus
• με το αυτοκίνητο = by car
• με το πλοίο = by ship / ferry

Greek often uses the definite article where English would not.

So Greek naturally says:

• με το τρένο = by train
• με το αυτοκίνητο = by car

Dropping the article is sometimes possible in certain contexts, but με το τρένο is the normal, natural wording here.

Yes. με normally takes the accusative.

The reason it may not look different is that τρένο is a neuter noun, and in the singular its nominative and accusative forms are the same:

• nominative: το τρένο
• accusative: το τρένο

So the form does not change, even though the case does.

Yes. Greek word order is more flexible than English.

This sentence could be rearranged, for example:

• Η μετακίνηση προς την Αθήνα είναι πιο εύκολη με το τρένο.
• Με το τρένο, η μετακίνηση προς την Αθήνα είναι πιο εύκολη.
• Η μετακίνηση με το τρένο προς την Αθήνα είναι πιο εύκολη.

These versions are all understandable, but the original sounds natural and neutral.

Changing the word order usually changes emphasis, not the basic meaning.

Yes, it is correct and natural, though it sounds a little neutral or slightly formal because of μετακίνηση and προς.

In everyday speech, many people might also say something like:

• Το να πας στην Αθήνα είναι πιο εύκολο με το τρένο.
• Είναι πιο εύκολο να πας στην Αθήνα με το τρένο.

These feel a bit more conversational.

So the original sentence is perfectly good Greek, but not the only natural way to express the idea.

Αθήνα is pronounced roughly a-THEE-na.

The noun itself does not change much between nominative and accusative:

• η Αθήνα
• την Αθήνα

Many feminine nouns in Greek do change only slightly, or not much visibly, in these forms.
The article is often the clearest sign of the case change:

• η = nominative
• την = accusative

So even though Αθήνα looks almost the same, the phrase as a whole clearly shows the case:

• προς την Αθήνα

Because this is not a full comparison like easier than X. It is a more general comparative statement.

Greek often uses πιο + adjective simply to mean more ... in context:

• πιο εύκολη = easier / more easy

If you wanted to say easier than traveling by car, then Greek would include από:

• Η μετακίνηση προς την Αθήνα είναι πιο εύκολη με το τρένο από ό,τι με το αυτοκίνητο.

So in the original sentence, than is not needed because the comparison is only implied.