I need to take a couple of days off for a job interview I need to fly to get there. What should I say at the "reason" part that won't make it too suspicious?
This is precisely the reason you should never give any more details than absolutely necessary when asking for time off. Just say "personal time" for every request you make, and then you don't need to lie in this scenario, because you just put "personal time" again.
The same advice as for every situation in life:
- Don't lie.
- Don't give away facts you don't want to give away.
- Use "I" liberally, but never "you".
This leaves you with the reason "taking care of personal/private matters".
Do not say "family matters". Looking for a new job is a personal matter, and "personal matter" is open enough to include almost everything else you could possibly do. "Family matters" would be a lie.
If someone keeps nagging, stick with it but not so that it gets aggressive or defensive. "Which personal matters?" - "Well, personal ones!". If they do not get the point after that flippant answer, your next escalation would be "I really do wish to keep this private." Be friendly while delivering that line. Do not say something like "I do not want to tell you" (whenever you use the "you" word you open yourself up for attack, or sound overly defensive - invoking the impression that you have something to hide).
For me, it would be inconceivable that someone keeps nagging after that. You can repeat the last line ad nauseam if they do (well, obviously not..., but you get the gist). If they keep asking then there is something else going on and you are beyond the scope of a cookie cutter answer like this; you'll have to find out how to weasel out of it yourself. But what you said before does not put you in any attackable position.
It may depend on the laws in your country. In germany the boss can simply deny you days off for a specific time period for a good reason. On the other hand he must give you days off if you need them for a job interview (even if you have used all vacation days for this year). In this case things get tricky and the only way seems to be telling him the real reason.