@zenvelo mentioned a 30 day notice, but in Canada we have to give 60 days’ notice. Whenever I’ve looked for a new apartment (I’m in apartment #5 now) I’ve always started looking 2 months before my move date. If you’re moving out of your parents’ place, you won’t have to give official notice, but keep in mind that when you’re moving out of an apartment you will have to give at least 30 days’ notice.
If you’re thinking of moving in May or June, you should be okay. It’s harder to find apartments for August 1st or September 1st, because of students coming into town, but May should be great (at least here, finals are sometime in April meaning that a lot of students vacate their apartments at the end of the month).
That being said, you likely won’t have a hard time finding a place. And I’m not sure what things are like where you’re from, but I’ve never not gotten an apartment that I applied for. As long as you don’t look like a crackhead and can pay the first and last up front, that should be enough.
Also, please follow the advice of others who’ve posted with respect to the building management, utilities etc. I’ve been fairly lucky myself, but I know people who’ve gotten into bad rental situations which could’ve been avoided had they done a little research!
Good luck!