It should be noted that the characteristic of a field is either prime or zero. If it is zero, then it contains the rational numbers. These are two statements you can probably prove even if algebra isn't your cup of tea.  



You can study valuation rings of mixed characteristic. A classic example is $\mathbb{Z}_p$ the p-adic integers. This is a ring of characteristic 0 and its fraction field $\mathbb{Q}_p$ is a field of characteristic 0. However, $\mathbb{Z}_p$ has a (unique) maximal ideal generated by $(p)\mathbb{Z}_p$ such that $$\mathbb{F}_p \cong \mathbb{Z}_p/(p)\mathbb{Z}_p$$ which is a finite field of characteristic $p$.  



There is the famous Ax-Kochen theorem, which is the following 

$$\Pi_\mathcal{F} \mathbb{Q}\_p \cong \Pi_\mathcal{F} \mathbb{F}_p((t))$$

where $\mathcal{F}$ is a non-principal ultra-filter on $\mathbb{N}$. This result  depends on the **continuum hypothesis.** (cf. However, there is also an isomorphism (J. Ax and S. Kochen, Diophantine problems over local fields I, American Journal of Mathematics,87 (1965), 605–630.)

However, there is also an isomorphism 

$$\Pi_\mathcal{F} \mathbb{Z}\_p = \Pi_\mathcal{F} \mathbb{F}_p[[t]]$$

where $\mathcal{F}$ is a non-principal ultrafilter on $\mathbb{N}$, which does not depend on the Continuum Hypothesis, in
"Use of Ultrapoducts in Commutative Algebra" by Hans Schoutens, found here
http://www.springer.com/mathematics/algebra/book/978-3-642-13367-1

*Witt Vectors* allow you to move from pure characteristic (either zero or positive) to mixed characteristic.

Another way of studying different characteristics is through Algebraic Geometry, by thinking of fields of different characteristics as fibers living over primes in $\mathbb{Z}$. 

Finally, there is even an "physical" way to interpret moving between fields. I highly recommend this article of A. Connes on the topic: 
"Characteristic one, entropy and the absolute point," Connes & Consani
http://arxiv.org/abs/0911.3537