I was a professional proofreader and editor a few years ago. It is challenging to start with but only a matter of time till you get into the swing of things.
My manager advised me to go through "slowly" the first time, correcting any errors, making changes to improve readability (for example, consistency of the terms used, abbreviations defined upon first use, an extra comma here or there, etc). Then, once finished, proofread again much more quickly. On the second proof, I found it helpful to zoom out (perhaps to 90% page size, as opposed to about 140% on the first read).
Use tracked changes, but untick the box where it says show ink etc. It is really helpful to only show comments. If you show ink whilst editing it can be easy to make errors (unnecessary spaces etc). It is much easier to have this switched off whilst editing. Then the author can switch them back on, view your changes and decide whether to accept/decline each one. Comments should be used if you are unclear of the meaning of a given sentence, and so cannot be sure that your correction reflects the author's meaning - you need to highlight this to them rather than just go ahead with what you think they mean.
Finally, as you proofread, it is really helpful to have a guide of the rules you are following. For example - numbers from 1-10 should be written in full, unless in a list containing number higher than 10 (e.g., 1, 4, 8, and 11 is correct) or for scores, ranges, etc. Beyond 10 (or some guides say 9), use the numerical form. There are a few things like this that need to be done consistently for a high quality paper. Have a Google and you should be able to find a guide for these sorts of things. But if not feel free to message me your email address and I will forward you one.
I hope this is helpful. Good luck :)