The day after tomorrow I _________ a game of tennis between 3 and 4, so come to see me after that.
This first sentence needs a bit of rearranging. Discussing such a small window of time so far into the future means that you're already talking about needing to do something on that particular day. Replying to their question, lead off with the day but still place the time after the event. For clarity's sake, you also want the time closer to your offer of availability afterwards.
Dealing with the verb 'have', Umonav's answer already covers the terminology pretty well. Something he didn't mention is that 'have' can be an action verb; it just almost exclusively concerns consuming food and beverages (I was having breakfast when suddenly..., e.g.) or planning (We'll be having the reunion over Memorial Day weekend, e.g.).
There are many ways to express future action. The differences are mostly about emphasis.
The standard way, which you mostly omitted, is to use will.
The day after tomorrow I will [be] play[ing] tennis between 3 and 4, so come and see me after that.
At that time in the future, I will do [or be in the middle of] this activity and you're not important enough for me to reschedule it instead of you. ['Go and ~', 'come and ~', &c. are more informal and technically correct formulations you sometimes see instead of 'go to ~', 'come to ~', &c.]
The day after tomorrow I shall [be] play[ing] tennis between 3 and 4, so come see me after that.
Some pedants like to retain the distinction between first-person 'shall' and second- and third-person 'will'. It's so uncommonly done these days that it probably even comes across as pompous even to the royal family. Unless your native language has trouble with terminal /l/ sounds (e.g. Mandarin-speakers) or your English teacher has a personal grudge against contractions, it's more informal and common to split the difference and use
The day after tomorrow I'll [be] play[ing] tennis between 3 and 4, so come see me after that.
which works for 'shall' or 'will'. ['Go ~', 'come ~', &c. are just asyndetic forms of 'go and ~', 'come ~', &c. and now more common than either other construction.]
The day after tomorrow I'm going to [be] play[ing] tennis between 3 and 4, so come see me after that.
This is more informal and variously emphasizes that informality; that the action is near term, as opposed to the immediacy of 'about to ~' or indefinite nature of 'will ~'; or that one is resolved that the action will occur in the manner 'shall' used to signify before everyone stopped using it in non-stilted speech.
The day after tomorrow I have a tennis match between 3 and 4, so come see me after that.
Pace userr..., this doesn't reflect a 'future of scheduling'. It's simply the present tense. I currently have this event scheduled, and you're not important enough to cancel or reschedule. ['Match' can imply a more serious contest than an informal 'game'; at the same time 'a noun of noun' is a more formal grammatical structure than 'attrib. noun'.]