I want to go up to 10 gauge strings on the Telecaster, will it play fine without doing any adjustments if I move up a gauge?
Maybe. Try it. If you want to change the gauge permanently, you might want to have some minor adjustments done.
Typically its fine if you do it, but if you want to keep it in tune as best as possible you will have to adjust the intonation of the guitar to your new gauges. No truss rod adjustment needed or crazy setup. So easy fix really, I agree with imaginary friend, you typically want to choose one string gauge and stick with it.