Theoretical/Empirical Arguments for Migrant Workers in the US
There is a broad literature that attempts to address the very question you pose- what is the impact of immigration (legal and illegal) on native wages? However, if you're interested in the 'real world' evidence, I'd suggest focusing on empirical labor economics as opposed to more theoretical labor models.

The reason for my suggestion is that theoretical economics makes no clear prediction without a lot more information. There are a variety of factors, such as the production function being considered, the labor market conditions observed, the relative substitutability of immigrant and native workers, etc. which tends to make answering the question more of an empirical exercise.

As far as the findings of the empirical papers, they actually tend to find relatively little impact of immigrant labor on native wages, with a few notable exceptions.

For some examples:

This survey paper, though a bit out of date as it's from 2005 outlines the results of 18 papers and finds wide discrepancies depending on modeling techniques and underlying assumptions used. 

Borjas, 2003 is one of the better known works, and finds what one might expect based on Econ101 models- that an increase in immigrant labor supply leads to a decrease in wages for native workers.

A couple very well known articles have tried to tease out the impact of immigration on wages by focusing on particular events (called supply shocks) where the number of immigrants increased "suddenly" for reasons having nothing to do with the economic climate at the time. A pair of papers examined the impact of the Mariel Boatlift, which led to a spike in Cuban immigrants in Miami: The first, a paper by Card 1990 found no evidence of a negative impact of immigration on wages, while Borjas 2015 finds a substantial decrease in the wages for the least educated Miami residents.

Several other papers have found either no negative effect, or even positive effects, of immigrants on various worker wages. For example, this paper by Peri in 2011 finds no negative impact of immigrants on wages, but a strongly positive impact of immigrants on productivity. This paper from 2013 finds significant positive impacts of undocumented immigrants on the wages of native workers in Georgia.

Ultimately, the exercise of determining the impact of immigrants on wages really depends on the method of analysis and a number of underlying parameters, and applying "standard" (i.e. Principles of Micro) models can lead to erroneous results and a false sense of certainty. 

If there's anything else in particular you're interested in, let me know and I'll try to improve the answer!

EDIT: Sorry, I forgot to include a link to this additional survey paper in the original answer!