As a PhD student there are limited funds available for travel to conferences. Should presenting at conferences with proceedings be prioritised over conferences without proceedings? Are conferences without proceedings considered less prestigious? I'm interested specifically in applied mechanics and materials science conferences.
I'm not familiar with your area, but I am strongly skeptical that it makes any sense to divide all conferences into two categories, those with proceedings and those without, and to declare based on that one binary bit of information that those in one category are more prestigious and more worth attending than those in the other category.
In other words, as with most things in life, the answer is it depends. Given two conferences you are considering choosing between, I would say look carefully at what topic each conference is about, how close is it to your interests, how good are the networking and learning opportunities you will get there, how attractive is the location, and yes, whether there is an opportunity to publish a proceedings paper; then make a decision.
In my experience, many PhD students tend to vastly underestimate the value of the networking that happens at conferences. At this stage in your career, your primary goal in going to conferences is to meet and become friendly with faculty at other institutions who are in your subfield. These are the people who will be in a position to give you a job after you graduate, and you will be in a much better position if they already know you when your job application arrives.
Of course, giving talks at conferences is also valuable for your CV, but I would say that this is of secondary importance. Of course, there are some fields (and I don't know specifically about applied mechanics or materials science) in which conference proceedings are the main route to publication, in which case it is vital to make sure that you go to at least enough conferences to publish all of your research. But you should certainly not limit yourself to conferences at which you will be able to publish.