I've written on this subject a few times, but the most relevant question ended up being closed, and the other is on Bio.SE, so I will mostly focus on tailoring the information there to the specifics of this question.
If you look at ether of my previous answers, you will note the first thing we need to establish is what is causing the common cold. From the body of evidence coming from the US and China I believe the answer to that is HRV followed by RSV.
Your immune system is actually directly attacked by two of the proteins RSV makes when it infects cells (NS1 and NS2). This knowledge is actually linked to a pretty cool vaccine idea for RSV. Thus I think it quite appropriate to think that active RSV infections weaken your immune system.
Looking at the very good epidemiology data coming from China (here's another example in addition to the first), it seems that the rate of co-infection, that is something else besides these two viruses infecting the patient, is quite high, particularity with RSV. This would imply that other pathogens take advantaged of a "weakened" immune system in an RSV infection.
It's less clear with HRV, but that might just be because of how amazingly common HRV infections are. It seems as though HRV induces a good immune response, in healthy patients, but it was still commonly found with other pathogens present (though not in rates as high as RSV). There is not a clear mechanism for HRV if it is in fact weakening the immune system (which actually might be better described as a "distracted" immune system in this case).
As for what you can do? Not a whole lot once symptoms show up. Contrary to whatever ads you may see, even some poor un-blinded trials, taking vitamin C is not going to help after you already have the cold. Even the prophylactic benefit seems to only be in certain cases, and I want to strongly discourage megadosing vitamin C, which in most cases does nothing helpful (and can cause problems of it's own).
Staying hydrated is always a good option, as is washing your hands and coughing/sneezing into your elbow to prevent infecting others. Other than that, you can hope for any of the vaccine candidates currently be researched to actually work, or that one of the small molecule inhibitors (anti-viral drugs) to actually be safe.