For integral weight $k$ and level $N=1$, I believe that the best result is due to Rankin (1990):
$$ \sum_{n\le X}a(n)\ll_\epsilon x^{1/3}(\log x)^{-\delta+\epsilon},
\qquad \delta:=\frac{8-3\sqrt{6}}{10}\approx 0.065.$$
On the other hand, Jutila (1987) proved that in square mean the sum is of size $\asymp x^{1/4}$, so the exponent $1/3$ cannot be lowered to $1/4$.
These results should generalize to arbitrary level and nebentypus. (The quoted book of Jutila (1987) proves the bound $\ll_\epsilon x^{1/3+\epsilon}$ with Voronoi summation, and this technique certainly generalizes. This bound is really due to Walfisz (1933), while the technique goes back to Wilton (1928), upon inserting the famous bound of Deligne (1974).)

For half-integral weight $k$, I don't know the best result from the top of my head, so I skip that part for the time being.

**Added.** Jutila's book is available online here. The quoted results are (1.5.21) and (1.5.23) on Page 28 (Page 30 in the original printed version).