Miután aláírom a bérleti szerződést, be akarok költözni.

Miután means after or once, but it introduces a full clause:

• Miután aláírom a bérleti szerződést... = After I sign the lease agreement...

By contrast, után is a postposition, so it comes after a noun or noun phrase:

• a bérleti szerződés után = after the lease agreement
• az aláírás után = after the signing

So a simple way to remember it is:

• miután + clause
• után + noun

Because Hungarian is using the definite conjugation here.

The object is a bérleti szerződést = the lease agreement, which is a definite object. When the object is definite, Hungarian often uses a different set of verb endings.

So:

• aláírok = I sign / I am signing something unspecified
• aláírom = I sign it / I sign the specific one

Compare:

• Aláírok egy szerződést. = I sign a contract.
• Aláírom a szerződést. = I sign the contract.

That -m ending in aláírom is the 1st person singular definite ending.

The -t marks the accusative case, which is usually the marker for the direct object in Hungarian.

Base form:

• szerződés = contract / agreement

Direct object form:

• szerződést = contract as the thing being signed

So in:

• aláírom a bérleti szerződést

the contract is what is being signed, so it gets the accusative ending.

Because Hungarian often uses the present tense for future time when the time reference is already clear from the context.

Here, miután already tells you the action happens later:

• Miután aláírom... = After I sign...

So Hungarian does not need a separate future form here.

This is very normal. English often does something similar in time clauses:

• After I sign the contract, ... not usually
• After I will sign the contract, ...

If you wanted to talk about a past event instead, you would use past tense:

• Miután aláírtam a bérleti szerződést, ... = After I signed / had signed the lease agreement, ...

Bérleti szerződés means lease agreement or rental contract.

It is made of:

• bérleti = rental / lease-related
• szerződés = contract / agreement

So literally it is something like rental agreement.

The form bérleti is an adjective. Hungarian often creates this kind of adjective to mean related to something.

Because be is a verbal prefix, and with a modal-like verb such as akarok (I want), that prefix often appears before the finite verb instead of staying attached to the infinitive.

So the basic verb is:

• beköltözni = to move in

But with akarok, Hungarian normally says:

• be akarok költözni = I want to move in

This is standard Hungarian word order.

A useful comparison:

• Beköltözöm. = I move in.
• Be akarok költözni. = I want to move in.

So the prefix has not changed meaning; it has just changed position.

It makes the meaning more specific.

• költözni = to move house / to relocate
• beköltözni = to move in
• kiköltözni = to move out
• elköltözni = to move away

So be gives the sense of movement into a place.

In this sentence, that is exactly what is meant: not just moving in general, but moving into the apartment/home after signing the lease.

Both are possible, but they are not equally neutral.

• be akarok költözni is the most natural, neutral way to say I want to move in
• beköltözni akarok can sound more emphatic or contrastive, as if stressing moving in specifically

So if you are just making a normal statement, be akarok költözni is the version to learn first.

Because Hungarian usually does not need subject pronouns when the verb ending already shows the subject.

• aláírom already means I sign
• akarok already means I want

So én (I) is usually omitted unless you want emphasis or contrast:

• Én akarok beköltözni, nem ő. = I want to move in, not him/her.

This is very common in Hungarian.

Because Miután aláírom a bérleti szerződést is a subordinate clause, and Hungarian normally separates subordinate clauses from the main clause with a comma.

So:

• Miután aláírom a bérleti szerződést, be akarok költözni.

This comma is standard punctuation.

Yes. That is a very natural alternative.

For example:

• A bérleti szerződés aláírása után be akarok költözni.

This means essentially the same thing: After signing the lease agreement, I want to move in.

The difference is mostly structural:

• Miután aláírom a bérleti szerződést... = clause-based, more like after I sign...
• A bérleti szerződés aláírása után... = noun-based, more like after the signing of the lease agreement...

Both are correct and common.

Yes, Hungarian word order is flexible, but changes in word order often change emphasis.

Neutral version:

• Miután aláírom a bérleti szerződést...

You could also say:

• Miután a bérleti szerződést aláírom...

but that puts more attention on the lease agreement or can sound more marked/stylized depending on context.

So for a learner, the original order is the safest and most neutral one to use.