Overview
There are standard approaches to answering this question that I feel do not satisfy those interested in quantitative details. So, in what follows I will explain the implications of the poloidal magnetic field using explicit solutions of the single-particle equations of motion, followed by plots of these solutions.
The motion of a particle in a toroidal field is composed of (1) a fast circular orbit of the particle about the magnetic field together with (2) slow parallel motion and drift motion of the center of the circular orbit in (1). The motion (1) is called the gyromotion and the motion (2) is call the parallel plus drift. Thus, in (2), we are talking about the motion of the gyrocenter. The notion of gyrocenter motion is fundamental to the theory of plasma transport, and is the basis of both neoclassical and gyrokinetic theory. Symbolically, the gyrocenter motion is written as
$$\mathbf{v} = v_\parallel \mathbf{b} + \mathbf{v}_d \; ,$$
where $v_\parallel = v_\parallel(\theta)$ is the parallel velocity, $\mathbf{b} = \mathbf{B}/B$ is a unit vector in the direction of the magnetic field $\mathbf{B}$, and $B = B(\theta)$ is the magnetic field strength. Also, $\mathbf{v}_d$ is the drift velocity. Drift velocity is discussed ad nauseum elsewhere and so we will just write the radial drift without a derivation.
Gyrocenter motion
The equations for gyrocenter motion are well-known, but for a tokamak with arbitrary shape they are complicated. 649
However, if the tokamak cross-section is circular, and the aspect ratio $R/r \gg 1$ (here, $R$ is the major radius and $r$ is the minor radius), then the equation are relatively simple. In this limit, the equations of motion are
\begin{align} \frac{d\varphi}{dt} \simeq &~ v_\parallel \mathbf{b} \cdot \mathbf{e}\varphi = \frac{v\parallel}{R_0} \quad \text{(toroidal direction)} \ \frac{d\theta}{dt} \simeq &~v_\parallel \mathbf{b} \cdot \mathbf{e}\theta =\frac{v\parallel}{q R_0} \quad \text{(poloidal direction)} \ \frac{dr}{dt} \simeq &~ \mathbf{v}d \cdot \mathbf{e}_r =-\frac{v\parallel^2+\mu B}{\Omega R_0} \sin\theta \quad \text{(radial direction)} \end{align}
I am using the fact that $v_\parallel \gg v_d$ to keep these equations as simple as possible. Note that the energy (over mass) $\varepsilon = v^2/2$ and magnetic moment $\mu = v_\perp^2/2B$ are conserved (constant along an orbit) quantities. Also, $\Omega$ is the gyrofrequency which depends on the particle mass (otherwise there is no explicit mass dependence). Finally, the safety factor ($q$) is the ratio of fields
$$q \simeq \frac{r B_\varphi}{R B_\theta}$$
This implies that in the limit of no poloidal field, $q \rightarrow \infty$. One interesting property of these orbits is that for some values of parameters, the orbits become trapped -- with a characteristic banana shape -- in the poloidal plane. These banana orbits are illustrated below.
Equations suitable for programming
As written, they are approximate but provide a huge amount of insight into the operation of a tokamak. However, they are still not quite in a useful form. According to plasma equilibrium theory, the magnetic field strength can be written as
$$B(\theta) = B_0 \left( 1-\epsilon \cos \theta\right) = B_0 G(\theta)$$
where $\epsilon = r/R \ll 1$. 649