I went to school for electrical engineering, and in my capstone electromagnetic theory course I encountered a problem that I needed to read papers to learn about. I asked my professor for his advice on how to read them, owing to my lack of expertise, and he recommended the following:
Don't read the abstract. This is written by experts for other experts, so they can quickly see if it is relevant to their work. It will provide no useful information, and may intimidate or confuse you.
Do read the introduction. This will contain an overview of experiment, and most importantly describe some of the background and motivation for the experiment. As a non-expert, this context is at a premium. If the introduction does not suggest there will be useful information, you can stop here.
Skip the body of the paper. The methods and construction of the work are usually not the goal of students, and this part is the most dense and requires the greatest expertise.
Do read the conclusion. This is usually the meat of what a student is looking for - what do experts think, in their own words, about their work. With this information in hand, you can decide whether you need to go deeper into the paper or have what you need. If you decide to go deeper:
Return to the body of the paper. This is especially useful if you are greatly surprised by the conclusion, or if you need to be able to reference images or graphs in an informed way. Even as a non-expert you can profitably think about how the conclusion follows from the steps they have laid out. It is naturally required if you want to attempt to duplicate the experiment.
I have had pretty good success with this method, even in subjects for which I have less than undergrad expertise. I will add two additional steps I found necessary, both obvious:
Reread the paper. As you follow up on the problem with additional reading, you will gain important new perspectives and context that led to missing things the first pass.
Follow-up the references. The assumptions about how diligent readers are with following through vary; this can make something aggravatingly opaque suddenly clear because it was covered well in the reference, or reveal a gaping hole the authors failed to address.
Anecdotally, the problem I was thinking about was the electrical signature of cancer. I had found a few modern references to work done on the subject in animals, but couldn't lay hold of work on humans for the life of me. Finally on the third pass over a survey paper covering related work I saw an obscure reference, and following through with that led me to all of the work which had been done on humans previously.