Is a surjective submersion between two manifolds always a fibration?

No. To see this, take a bona fide fibration such as $\operatorname{pr}_2:\mathbf{R}^2\to\mathbf{R}$ and remove a point from the domain. Meigniez on p. 3778 lists a number of sufficient conditions that a submersion $f$ be a fibration. The best known, already noted by Damian and Donu, is that $f$ be proper as in Ehresmann's Theorem (proved e.g. in Bröcker and Jänich, (8.12)). 


Suppose that $f:X\rightarrow Y$ is a surjective submersion of manifolds. Let $n=\dim(X)$ and $m=\dim(Y)$. Suppose that $y\in Y$. In suitable local coordinates $(x_1,\ldots,x_n)$ on $X$ and $(y_1,\ldots,y_m)$ on $Y$ at $y$, $f$ has the form $(a_1,\ldots,a_n)\mapsto (a_1,\ldots,a_m)$. Hence, you should obtain a nice $(nm)$dimensional fibre. This should be made rigorous, but I think it gives a decent place to start. 

