Let $(X_t)_{t \geq 0}$ be a solution to the SDE
$$dX_t = \sigma(X_t) \, dW_t + b(X_t) \, dt, \qquad X_0 =x. \tag{1}$$
Throughout my answer, I will assume that the drift coefficient $b$ and the diffusion coefficient $\sigma$ are "nice", e.g. nice enough to ensure the existence of a unique solution to the SDE.
As you already mentioned, $b$ and $\sigma$ play an important role for the study of moments of $X_t$. The drift coefficients determines $\mathbb{E}(X_t)$ whereas $\sigma$ characterizes, via the Itô isometry, the variance $\mathbb{E}((X_t-\mathbb{E}(X_t))^2)$. In particular,
$$|\mathbb{E}(X_t)| \leq \|b\|_{\infty} t \qquad \text{and} \qquad \text{var} (X_t) \leq \|\sigma^2\|_{\infty} t.$$
It is possible to obtain bounds for moments $\mathbb{E}f(X_t)$ for a much larger class of functions $f$, e.g. to study exponential moments of $X_t$. Typically the bounds look similar as for Brownian motion; however, they involve the supremum norms $\|b\|_{\infty}$ and $\|\sigma^2\|_{\infty}$. If the coefficients are unbounded, then it is more reasonable to study $\mathbb{E}f(X_{t \wedge \tau_r})$ where $$\tau_r := \inf\{t \geq 0; |X_t| \geq r\};$$ the bounds for these moments then will involve $\sup_{|x| \leq r} |b(x)|$ and $\sup_{|x| \leq r} |\sigma(x)^2|$.
If we want to get more information on the solution, then a possible approach is to take a look at the transition density of the process. There is a nice result by Fournier & Printems $\{3\}$ which gives a sufficient condition for the existence of the density.
Theorem: If $\sigma$ is Hölder continuous with exponent $\alpha \in (1/2,1]$ and $b$ grows at most linearly, then $X_t$ has a density on the set $\{x; \sigma(x) \neq 0\}$.
There is a quite extensive literature on heat kernel estimates for the transition density, see e.g. Aronson $\{1\}$ for a classical result. Using the heat kernel estimates, it is possible to deduce lots of information on the properties of $(X_t)_{t \geq 0}$.
A second line of approach is to study the solution $(X_t)_{t \geq 0}$ as a Markov process. Since $(X_t)_{t \geq 0}$ is Markovian, we can associate a so-called infinitesimal generator $A$ with the process. Roughly speaking, $\mathbb{E}^x f(X_t)-f(x) \approx t Af(x)$ for small $t$, see this question for some more information. The generator of the SDE $(1)$ is of a very particular form; for any function $f$ which is smooth and has compact support it holds that
$$Af(x) = b(x) f'(x) + \frac{1}{2} \sigma(x)^2 f''(x).$$
Alternatively, we can represent $A$ as a pseudo-differential operator
$$Af(x) = - \int_{\mathbb{R}} q(x,\xi) e^{ix \xi} \hat{f}(\xi) \, d\xi$$
where $\hat{f}$ is the Fourier transform of $f$ and
$$q(x,\xi) := -ib(x) \xi + \frac{\sigma^2(x)}{2} \xi^2, \qquad x,\xi \in \mathbb{R}$$
is the so-called symbol. This allows us to apply results which are known for Feller processes (a class of nice Markov process), cf. $\{2\}$. For instance we can obtain the following maximal inequality
Theorem: There exists a constant $c>0$ (not depending on $b$ and $\sigma$) such that $$\mathbb{P}^x \left( \sup_{s \leq t} |X_s-x|>r \right) \leq ct \sup_{|y-x| \leq r} \sup_{|\xi| \leq r^{-1}} |q(y,\xi)| \quad \text{for all $t \geq 0$}. \tag{2}$$
Note that the left-hand side of $(2)$ tells us how fast the process $(X_t)_{t \geq 0}$ is moving away from its starting point $x$, that is, if the probability is small, then $(X_t)_{t \geq 0}$ stays (at least for small times $t$) close to its starting point. In fact, it is possible to show that
$$\limsup_{t \to 0} \frac{1}{t} \mathbb{P} \left( \sup_{s \leq t} |X_s-x| \geq r \right) \leq C \sup_{|\xi| \leq r^{-1}} |q(x,\xi)| \tag{3}$$
which is a really helpful result if one is interested in the small time asymptotics of $(X_t)_{t \geq 0}$. If $\sigma(x) \neq 0$, then the diffusion part will dominate the small time behaviour of $(X_t)_{t \geq 0}$,
$$\limsup_{t \to 0} \frac{1}{t} \mathbb{P} \left( \sup_{s \leq t} |X_s-x| \geq r \right) \leq C \sigma(x) r^{-2}.$$
Accordingly it is possible to obtain bounds for the growth of the sample paths, e.g. if $\sigma>0$ then the sample paths of $(X_t)_{t \geq 0}$ will behave for small times similar as the ones of a Brownian motion, see e.g. Section 5.3 in for some results in $\{2\}$ this direction.
$\{1\}$ Aronson, D.G.: Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 (1967), 890-896.
$\{2\}$ Böttcher, B., Schilling, R.L., Wang, J.: Lévy Matters III. Springer, 2013.
$\{3\}$ Fournier, N., Printems, J.: Absolute continuity for some one-dimensional processes. Bernoulli 16 (2010), 343-360.
