Learn Hardcore Finnish: Minä haluan pelata tennistä ystäväni kanssa.

Minä haluan pelata tennistä… – perfectly correct, slightly more emphatic on I.
Haluan pelata tennistä… – also perfectly correct, and very natural.

Use Minä when you want to contrast or emphasize:
Minä haluan pelata tennistä, mutta hän haluaa pelata jalkapalloa.
“I want to play tennis, but he/she wants to play football.”

Why is it haluan and not just halua?

Haluan is haluta (“to want”) conjugated for 1st person singular, present tense:

infinitive (dictionary form): haluta – to want
stem: halua-
1st person singular ending: -n

So: halua- + n → haluan = I want.

Similarly:

(minä) puhun – I speak
(minä) syön – I eat

Why is pelata not conjugated (like pelaan)? Why “haluan pelata” and not “haluan pelaan”?

In Finnish, when one verb expresses wanting / being able / having to etc., the second verb usually stays in the basic infinitive form.

The pattern is:

[conjugated verb] + [infinitive]

So:

haluan pelata – I want to play
voin pelata – I can play
aion pelata – I’m going to play / I intend to play
haluaisin pelata – I would like to play

You do not say *haluan pelaan. Only the first verb takes the personal ending, the second stays pelata.

Why is it tennistä instead of just tennis?

Tennistä is tennis in the partitive singular case.

Here, tennistä is the object of pelata, and for playing sports and games Finnish typically uses the partitive case to show an activity that is:

ongoing / uncompleted, or
of unspecified amount/duration.

So:

pelata tennistä – to play (some) tennis
pelata jalkapalloa – to play (some) football
pelata shakkia – to play chess

You cannot leave the word in bare dictionary form; in a real sentence it needs a case ending. So *pelata tennis is ungrammatical; it must be pelata tennistä.

Finnish has no articles like a or the. How do I know if tennistä means “tennis”, “some tennis”, “the tennis”, etc.?

Finnish simply doesn’t mark the difference between a, some, and the with separate words. Context does that job.

Haluan pelata tennistä.
Could be translated as “I want to play tennis”, “I want to play some tennis”, or even “I want to play the tennis (match)” depending on context.

Learners usually translate tennistä as “tennis” or “some tennis” in neutral contexts. There is no separate article word to worry about.

What exactly does ystäväni mean? How is it formed?

Base noun: ystävä – friend.

Add the possessive suffix -ni to indicate “my”:

ystävä + ni → ystäväni – my friend

So ystäväni = friend-my = my friend.

This suffix can also create ambiguity in other contexts (it can look like a plural), but here, with kanssa, it clearly means “with my friend”.

Can I also say minun ystäväni kanssa? Is that different from ystäväni kanssa?

Yes, you can say both:

ystäväni kanssa – with my friend
minun ystäväni kanssa – with my friend (a bit more explicit/emphatic)

Details:

-ni on ystäväni already means “my”.
Adding minun makes the possession more explicit or emphatic, similar to stressing my in English.

Both are correct. In everyday speech, ystäväni kanssa alone is very common and perfectly natural.

Why is it ystäväni kanssa and not *kanssa ystäväni? Can the word order change?

Kanssa is a postposition, not a preposition. That means it normally comes after the noun:

ystäväni kanssa – with my friend
äitini kanssa – with my mother
opettajan kanssa – with the teacher

The usual order is:

[noun / pronoun] + kanssa

So ystäväni kanssa is the default.
You can move the whole phrase in the sentence for emphasis:

Haluan pelata tennistä ystäväni kanssa. (neutral)
Ystäväni kanssa haluan pelata tennistä. (emphasis on with my friend)

But kanssa still stays after the noun it belongs to.

Why does ystäväni look like it has no -n genitive ending, even though kanssa usually takes the genitive?

Without a possessive suffix:

ystävä (nominative) – friend
ystävän (genitive) – friend’s / of (a) friend
ystävän kanssa – with a friend

When you add the possessive suffix -ni, the -n of the genitive is “absorbed” into the suffix:

ystävä + n + ni → ystäväni (spelled just ystäväni)

So ystäväni here is actually genitive singular + “my”:

grammatically: ystäväni (GEN.SG + poss. -ni)
meaning: my friend (required by kanssa)

That’s why you don’t see a separate -n before -ni.

Can I change the word order in the whole sentence? For example, is Haluan ystäväni kanssa pelata tennistä okay?

Yes, word order in Finnish is relatively flexible, as long as relationships between words (cases, endings) are clear.

All of these are grammatical, with slight nuances in emphasis:

Haluan pelata tennistä ystäväni kanssa. (neutral, very natural)
Haluan ystäväni kanssa pelata tennistä. (slight emphasis on with my friend)
Ystäväni kanssa haluan pelata tennistä. (stronger focus on with my friend)

The most basic, textbook word order is the first one:
(Minä) haluan pelata tennistä ystäväni kanssa.

How do I say the negative: “I don’t want to play tennis with my friend”?

Negation in Finnish uses a special negative verb ei, which is conjugated, and the main verb goes into a special form (the “connegative”):

haluta → halua (for the negative)
1st person singular of ei is en

So:

En halua pelata tennistä ystäväni kanssa.
= “I don’t want to play tennis with my friend.”

Structure:

en (I don’t) + halua (want) + pelata tennistä ystäväni kanssa (to play tennis with my friend).