Learn Hardcore Hungarian: Ott veszek jegyet a vonatra péntek reggel. - There I buy a ticket for the train on Friday morning.

Ott veszek jegyet… – literally: I buy a ticket there…
Because of péntek reggel (Friday morning), it is clearly about a future time, so English needs will.

You can also use the explicit future with fog:

Ott fogok jegyet venni a vonatra péntek reggel. – I will buy a ticket there for the train on Friday morning.

Both are correct; the simple present is very common and neutral when a time is given.

Why is it jegyet and not jegy?

Jegyet is the accusative form of jegy (ticket). Hungarian marks the direct object with -t:

jegy – ticket (dictionary form)
jegyet – ticket as a direct object (a ticket)

So:

Veszek jegyet. – I buy a ticket.
Veszem a jegyet. – I am buying the ticket.

In the sentence Ott veszek jegyet…, you are talking about a ticket (indefinite object), so you use jegyet and the indefinite verb form veszek.

Why is it veszek and not veszem?

Hungarian verbs have two main conjugations:

indefinite (when the object is indefinite: a ticket, something)
definite (when the object is definite: the ticket, it)

Examples:

Veszek jegyet. – I buy a ticket. (indefinite object → veszek)
Veszem a jegyet. – I buy the ticket. (definite object → veszem)

In Ott veszek jegyet a vonatra…, the key object is just a ticket, not a specific known ticket, so veszek is correct.

What does the ending -ra in vonatra do? Why not just vonat?

Vonat means train. The suffix -ra/-re usually means onto / to (a surface or goal). In this context it is used in a more abstract sense: onto the train = for the train.

vonat – train
vonatra – onto the train / for the train

With jegyet venni (to buy a ticket), Hungarian normally says jegyet venni valamire (to buy a ticket for something):

jegyet venni a buszra – to buy a ticket for the bus
jegyet venni a vonatra – to buy a ticket for the train

So jegyet a vonatra literally is ticket onto the train, understood as ticket for the train.

Can I say vonatjegyet instead of jegyet a vonatra?

Yes, you can:

Ott veszek vonatjegyet péntek reggel.

Vonatjegy is a compound noun meaning train ticket. The difference:

jegyet a vonatra – more literal, ticket for the train
vonatjegyet – a set phrase, sounds very natural and compact

Both are correct and common. In everyday speech, vonatjegyet is probably more frequent.

Could the word order be different, like Péntek reggel ott veszek jegyet a vonatra?

Yes. Some natural variants (all correct, slightly different emphasis):

Ott veszek jegyet a vonatra péntek reggel.
Péntek reggel ott veszek jegyet a vonatra.
Péntek reggel jegyet veszek a vonatra ott. – possible, but ott sounds tacked on at the end; less typical.

Neutral, clear options are the first two. Hungarian word order mainly serves to show topic and focus, not just subject–verb–object. Here:

Putting Ott first highlights the place.
Putting Péntek reggel first highlights the time.

What is the role of ott, and how is it different from itt or oda?

ott – there (static place: you are talking about some place that is not where you currently are)
itt – here (where the speaker is)
oda – (to) there, in the sense of movement/direction

In the sentence:

Ott veszek jegyet… – I will buy a ticket there (at that place).

If you wanted to emphasize movement, you could say for example:

Oda megyek, és ott veszek jegyet. – I’ll go there and buy a ticket there.

Why is it péntek reggel and not something like on Friday morning with a separate word?

Hungarian usually does not use a preposition like on for days and parts of the day. Instead:

péntek reggel – Friday morning / on Friday morning
hétfő este – Monday evening / on Monday evening

You can also say:

pénteken reggel – also correct, a bit more explicit/literary

So:

Ott veszek jegyet a vonatra péntek reggel.
Ott veszek jegyet a vonatra pénteken reggel.

Both mean: I’ll buy a ticket for the train there on Friday morning.

Where would nem (not) go if I want to say I will not buy a ticket there?

In neutral sentences, nem goes right before the verb (or the focused element + verb). So:

Ott nem veszek jegyet a vonatra péntek reggel.
– I will not buy a ticket for the train there on Friday morning.

Other possible word orders (still correct):

Péntek reggel nem veszek jegyet a vonatra ott.
Péntek reggel ott nem veszek jegyet a vonatra.

But the clearest and most neutral negation for your original sentence is:

Ott nem veszek jegyet a vonatra péntek reggel.

How can I make the sentence more explicitly future or more polite?

For a more explicit future, use fog:

Ott fogok jegyet venni a vonatra péntek reggel.
– I will buy a ticket for the train there on Friday morning.

For polite/softened intention, you can use szeretnék (I would like):

Ott szeretnék jegyet venni a vonatra péntek reggel.
– I would like to buy a ticket for the train there on Friday morning.

Both are very natural, especially when you’re speaking to someone you don’t know well (e.g. planning, asking, or explaining your intentions).