Is there a model of computation, that tries to be realistic? For instance, the tape on a Turing machine is infinite, where as we usually only have a finite amount of available memory.  Secondly Turing machines are not really convenient IMHO for proving things about everyday algorithms on an everyday computer.

I want a model that makes computations in a ring, namely $\Bbb{Z}_p\times \dots \times \Bbb{Z}_p = (\Bbb{Z}_p)^k$, where $p$ is prime (for a good reason... later on that).  Since that's what bytes on a computer look like they look like values in $\Bbb{Z}_8$ and even some of the instructions on those bytes on modern computers cause wrap around that looks like that of $\Bbb{Z}_8$.  It's obvious to me that any computation on my home PC can be done with computations in said ring.  But of course I'd have to prove that formally if I wrote a paper.  I can already write conditional assignments of a memory slot (element of $\Bbb{Z}_p$), as the value of a polynomial in $\Bbb{Z}_p[x_1, \dots, x_k]$.

It's hard to find literature on this, but to me it seems so obvious that this is a nice approach and I'm shocked that no one's considered this yet.  Or have they?  Links please...
As D.W. mentions in the comments, the model you are looking for is the RAM machine. In fact, there are several different models, and I recommend the following recent paper of Fürer's for a good overview.

In contrast to what you describe, the correct ring is $\mathbb{Z}_{2^n}$, where $n$ is the width of registers – up to 512 bits in future Intel processors. The exact value of $n$ doesn't matter at all as long as it's constant, since in computational complexity we are always interested in time and space complexity up to a multiplicative constant. Therefore, to make matters interesting, the RAM models described above let $n$ depend on the input size.

Regarding finitude of memory, this is captured, to some extent, by space constraints – polynomial space (or linear/quasilinear space) for algorithms working on normal data, logarithmic space (or polylogarithmic space) for algorithms working on big data; there are also streaming models for even less space. Restricting the space altogether would be destructive since then all deterministic algorithms would terminate in constant time – to make sense of asymptotics, we must allow space to grow with input size.