Asked out this girl. She told me she has plans. I teased her about cancelling plans and going out with me. She said next time. Should I suggest a new date right away or just ask her again later?

Hey, Julie, I'd love to go to the movies with you. Let me know when you're free - maybe, this weekend?

If you want her to break a date with someone just to go out with you, be prepared to have the same thing happen to you. That's just bad juju, if you cannot wait for someone to be free.

I might wait a bit and ask her again.  It all depends if you think she really has plans or might not be interested in going out with you. 

"Next time" as her reply indicates to me that while the issues is closed for the moment (for any one of a number of positive reasons that do not involve you at all), the next time would be a brand new situation.

I would definitely ask her again, but not right away.

Danae Hitch's suggestion works for older people  (20's or so).

The problem with doing that if you are younger is that you are really forcing here to determine when the date is going to take place (even though that is not your intent).

She might come to resent that.