I am currently using an F15T8NDL 18" lightbulb for my plants in a light fixture, and would like to replace it with something much brighter. So my question is that if I add a higher powered bulb in the same fixture, will it automatically draw more power and be much brighter, or do I need a new fixture? And if so, what would be a good replacement bulb?
If you want more light from the same florescent fixture, you will need a new ballast (power supply from "wall power" to "bulb power") as well as a new bulb. At which point a whole new fixture might make more sense.
Not having fiddled with 18" bulbs any time recently I'll leave the question of exactly what you might replace it with to others.
These days you should compare LED options with florescent options and possibly also look at HID bulbs as well, comparing all on a light out to power in basis (though "light useful to plants out to power in" is harder to compare easily with most published information.)