I'd like to know an upper bound for $\zeta(s)$ in the critical strip, and hopefully one that is not too difficult to prove.
For instance, http://math.univ-lille1.fr/~ramare/TME-EMT/Articles/Art06.html cites a 1918 result that $\zeta(s) \ll t^{(1-\sigma)/2} \log t$. I don't see how to prove this, and in fact the factor $\frac{1}{2}$ in the exponent is surprising to me, based on what you would get from the Hardy-Littlewood approximation.