The 15 inch tires on the Corolla are 195/65 R15. Going from 15s to 16s the equivalent size is 205/55 R16 (that way the speedometer displacement is less than 0.00001% -- meaning your speedometer doesn't need to be re-calibrated). Also the suspension doesn't have to take a lot of punishment if you're using 16s with that size.
For 17s the equivalent tire size would be 225/45 R17 or 255/40 R17.
And for 18s the equivalent tire size would be 255/35 R18 or 295/30 R18.
I wold HIGHLY advise against going for 18s. And I would highly recommend not going above 16s.