Vergil I wouldn't recommend too soon, but Ovid I would. The only thing is that grammars don't really prepare you for poetry and the word order will seem... insane. But you get used to that!
If you intend to read poetry it is really useful to get some practice at writing in metre yourself, it greatly increases your appreciation for the art. WhiteOctave has written a great piece on latin elegiac meter, which I understand is what Ovid almost always used (i believe with the exception of Metamorphosis or Heroids - not sure which one).
However, regardless of what you start with, you're going to have to learn vocabulary. Every writer has his own and this will make the reading very slow at first. I've been studying latin for about a year and half now and I've just started on Ovid: it takes me about half an hour to read only 50 lines! However, slowly it is getting quicker and I'm enjoying it more.
A grammar only opens a door, then there is a long walk ahead.
“Cum ego verbo utar,” Humpty Dumpty dixit voce contempta, “indicat illud quod optem – nec plus nec minus.”
“Est tamen rogatio” dixit Alice, “an efficere verba tot res indicare possis.”
“Rogatio est, “Humpty Dumpty responsit, “quae fiat magister – id cunctum est.”