После работы я поеду домой на машине.

Because после requires the genitive case.

• Dictionary form: работа
• Genitive singular: работы

So после работы literally means after work.

This is a very common pattern in Russian:

• после урока — after the lesson
• после обеда — after lunch
• после фильма — after the film

Поеду is used for one specific trip in the future.

• ехать / еду = to be going, to be traveling by vehicle
• поехать / поеду = to go, to set off by vehicle, for a single completed trip

In this sentence, the speaker means after work, I will go home by car, so a future, one-time action fits поеду well.

By contrast:

• еду домой = I am going home / I’m on my way home
• поеду домой = I will go home

Russian often shows the future inside the verb itself, so it does not need a separate word like English will.

Here, поеду already means I will go.

That happens because поехать is a perfective verb. Perfective verbs form the future with simple personal endings:

• я поеду — I will go
• ты поедешь — you will go
• он поедет — he will go

So поеду already contains the idea of will.

Because Russian chooses different motion verbs depending on how you move.

• идти / пойти = go on foot
• ехать / поехать = go by vehicle

Since the sentence says на машине — by car — the correct verb is поеду, not пойду.

Compare:

• Я пойду домой. — I’ll go home on foot.
• Я поеду домой. — I’ll go home by vehicle.

The prefix по- turns ехать into поехать, which is the perfective verb.

In this kind of motion verb, по- often gives the sense of:

• setting off
• starting a trip
• a single whole trip

So поеду is not just I go; it is more like I will go / I will set off.

That is why it works naturally for a planned future action.

Because домой means homeward / to home.

These forms are different:

• домой = to home, homeward
• дома = at home
• к дому = toward the house/building

So:

• Я иду домой. — I’m going home.
• Я дома. — I’m at home.
• Я подхожу к дому. — I’m approaching the house.

In your sentence, the destination is home, so домой is the right choice.

Because домой is an adverb, not a noun phrase.

English uses to in go home only in older or more formal styles, but modern English usually just says go home. Russian works similarly here: домой already contains the idea of direction, so no extra preposition is needed.

That is why Russian says:

• идти домой
• ехать домой

not в домой.

In most cases, домой naturally means to one’s home, so Russian usually does not need to add мой.

Russian often leaves this understood from context.

So:

• Я еду домой normally means I’m going home
• not necessarily I’m going to a house

If you really need to stress my own home, you can say к себе домой or sometimes в свой дом, but that is much more specific and less neutral.

На машине means by car.

Here, машине is in the prepositional case after на. Russian very often uses this pattern for transport:

• на машине — by car
• на автобусе — by bus
• на поезде — by train
• на самолёте — by plane

This is just the normal Russian way to express means of transport in many cases.

It is not best understood literally as on the car. It is simply the standard idiomatic pattern.

It can be omitted if the subject is clear.

Russian verb endings often already show the person:

• поеду = I will go

So После работы поеду домой на машине is perfectly natural in context.

Including я is also fine. It may sound:

• a little clearer
• slightly more explicit
• sometimes more contrastive or emphatic

So both are possible:

• После работы я поеду домой на машине.
• После работы поеду домой на машине.

Yes. Russian word order is flexible, and changing it changes the focus or emphasis, not the basic meaning.

The original sentence is neutral:

• После работы я поеду домой на машине.

Other possible orders:

• Я после работы поеду домой на машине.
• Домой я поеду после работы.
• На машине я поеду домой после работы.

All of these can work, but they highlight different parts of the sentence. Russian often puts the most important or contrastive information in a more prominent position.

Then you would normally use ездить, not поехать.

• После работы я езжу домой на машине. = After work, I go home by car / I usually go home by car.

Why?

• поеду = one future trip
• езжу = repeated, habitual action

This is an important motion-verb contrast in Russian:

• ехать / поехать = one trip in progress or one specific trip
• ездить = repeated trips, habit, ability, general fact