Odotan vartin ja sitten menen kotiin.

Finnish often drops subject pronouns because the verb ending already shows the person.

• Odotan = I wait / I’m waiting (1st person singular)
• menen = I go / I’m going (1st person singular)
  You can add minä for emphasis or contrast: Minä odotan vartin… (implying I will wait, maybe others won’t).

In odotan, the ending -n marks 1st person singular in the present tense. The dictionary form is odottaa (to wait).
Conjugation pattern here:

• minä odotan = I wait
• sinä odotat = you wait
• hän odottaa = he/she waits

Vartti means a quarter of an hour (15 minutes).
Both forms can occur, but they feel different:

• Odotan vartin.
  Uses the genitive/“accusative-like” form vartin to express a bounded, complete amount of time: I’ll wait (for) 15 minutes (and then stop).

• Odotan varttia.
  Uses the partitive varttia, which often feels more like I’ll wait for 15 minutes / I’ll be waiting for 15 minutes with a slightly more “ongoing” feel (and in some contexts can sound less “neatly bounded”).

In everyday Finnish, odottaa commonly takes the partitive for objects (odotan bussia = I’m waiting for the bus), but time expressions like this often allow the bounded -n form when the duration is seen as a complete unit.

Vartti is very common and neutral in spoken and informal written Finnish. It’s essentially the everyday word for 15 minutes. In more formal contexts you might also see:

• viisitoista minuuttia = fifteen minutes
  But vartti is not “slangy” in a problematic way—just more conversational.

Kotiin is the illative form of koti and means into/to (one’s) home—it expresses direction toward the inside of a place.

Compare:

• kotiin = to home (toward/into home)
• kotona = at home (location)
• kodista = from home (out of home)

So menen kotiin specifically encodes going home (to the home) rather than just being home.

Sitten means then and marks the next step in a sequence.
The structure … ja sitten … is very common: it clearly signals first X, and then Y.

You can move it for emphasis, but the most natural is exactly what you have:

• Odotan vartin ja sitten menen kotiin.

Alternatives are possible, with slight changes in rhythm/emphasis:

• Odotan vartin, sitten menen kotiin. (still very natural)
• Sitten menen kotiin. (as a standalone next-step sentence)

ja means and and connects two actions into one smooth plan: I’ll wait 15 minutes and then I’ll go home.
You can omit ja and use punctuation instead, especially in writing:

• Odotan vartin, sitten menen kotiin.
  Both are correct; ja sitten just feels especially “flowing” and conversational.

Finnish word order is fairly flexible, but changes often affect emphasis.

Neutral, natural:

• Odotan vartin ja sitten menen kotiin.

If you front kotiin, it emphasizes home as the destination:

• Kotiin menen sitten vartin odotettuani. (more literary/structured)

If you front the time, it emphasizes the waiting duration:

• Vartin odotan ja sitten menen kotiin. (a bit marked, but possible)

The original sentence is the most straightforward everyday version.