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I've submitted my DNA to Ancestry and GEDmatch in a desperate attempt to match to anyone from my paternal side.
I am unsure of who my father is and my mother will take that information to her grave. I really need help because I am new to this.
On Ancestry, the ones with the closest relationship were cousins. I already messaged them, but they were from my maternal side.
legacyfamilytreewebinars.com ran two fairly recent webinars that might be extremely useful to you, "Weaving DNA Test Results into a Proof Argument" by Karen Stanbury, and "Finding Missing Persons With DNA Testing", by Diahan Southard. You would need to buy at least a short-term subscription to the webinar site to view them, but in your situation, I suspect you'd find it worthwhile (I have no affiliation with Legacy Family Tree or the webinar site).
I think you've started out on the right path by (a) testing at AncestryDNA, the largest DNA subscriber base around, and (b) submitting your sample data to GEDmatch.com. @Jaclyn is correct that the next step is to find close maternal relatives to get tested, so that you can use the details of those matches to filter out more distant maternal-side matches. In practice, it will take more than one, possibly many. Obviously, the best option would be to get your mother tested, but from what you say that seems unlikely. The next best choice would be to essentially try to reconstruct her genome (or that of her close ancestors) using first your own siblings (if any), then your mother's siblings (if any), first cousins, etc.
Using the chromosome browser at GEDmatch, you can compare your sample to your close maternal relatives and identify specific DNA segments that map to your maternal ancestors.
At that point, what you're looking for is "shared matches" on AncestryDNA who match you, but not any of your maternal relatives. When you find some, you may need to persuade them to also upload their sample data to GEDmatch. If the segment(s) you match overlap with segments that you share with maternal relatives, but they don't match those relatives, then most likely they're matching on the other half of the half-IBD (i.e., your paternal DNA).
If, on the other hand, they match segments that you haven't established maternal-side matches for, you can't be certain, unless you have enough samples among your maternal relatives that the lack of a "shared match" with any of them indicates a high statistical likelihood of a paternal-side match.
With enough such paternal-side matches, you may be able to form a hypothesis as to a most recent common paternal ancestor, where your paternal matches fit in that tree and, using the ISOGG DNA relationship tables, where your father might fit. You can then run your theory past your mother and see how nervous she gets.
The DNADetectives Facebook group is also useful, but be aware that, like most such groups, there's a tendency to over-emphasize the success stories, which are actually the far end of the bell curve. The likelihood of finding a close paternal cousin match on any of the commercial DNA sites is statistically fairly small (yes, a few people do, but not many). What you're much more likely to see is matches against somewhat more distant cousins, but more of them (possible many more).
If you order a 23andme kit for yourself and can get one of your first cousins or a sibling of your mom to also submit a 23andme kit, you can deduce which cousin matches might be related to you through your father's side by triangulating which cousin matches are from your mother's side. Also if you are male, have a look at your ydna / paternal haplogroup, which you inherited from your dad, and see which of the cousin matches have the same ydna to identify which ones are your paternal cousins.
Additionally, you can use FamilyTreeDNA's ydna tests to get more precise in your paternal research. With the FTDNA ydna tests with higher markers, you can find out which surnames are the most common to help you narrow down the identity of your father.
Lastly, there is a great Facebook group called "DNA Detectives," which is a community of people who are passionate about genetic genealogy and assisting adoptees and other people searching for their biological relatives. They can give further guidance how to identify which matches are from your paternal side as you conduct your search.
Your question about how to find DNA matches from the paternal side is a very good one, and the techniques used to do that are not simple and there are several other answers that give some clues as to doing that.
But I keep looking at your question over and over and can't bring myself to believe that you have left out two more important questions that you strongly suggest but don't ask.
You say your mother will take "to her grave" the information about who your father is. That very strongly suggests that she does not want you to know. So the first question it seems you are asking is: "Should I go behind my mother's back and look for who my father is?". This is a very serious question and if and when your mother finds out you are doing this, it may ruin whatever relationship you have for the remainder of your lives. Is this important enough for you to do this and risk losing your mother? Or can you wait until after she's gone?
Should you decide you're doing this anyway, no matter the consequences, then this is obviously extremely important to you. In cases of adoption or unknown parents, you shouldn't be asking how to determine paternal side of DNA. You should be asking "How do I find out who my father is?". In cases like that, I would recommend seeking a specialist in finding parents through DNA. There are many excellent people you can contact (I don't want to show preferential treatment to any here), but you can find some from the Association of Professional Genealogists site on their search page by Research Specialty with a search for "Adoption" or "DNA specialist". The good services are starting to get very high success rates and can help you greatly and will know exactly how to pinpoint who your father is and would be able to do so much quicker than you could yourself. Yes, it costs, but so does taking DNA tests yourself and how important is this to you? Not only that, but there will be less chance of your mother finding out you are doing this if a 3rd party does this discreetly for you as you may ask them to.
If I've got this wrong and the intent of the question is really to figure out which DNA matches are from the paternal side, then please remove the line "I am unsure of who my father is and my mother will take that information to her grave." from your question and then get a moderator to remove this answer that I've posted.
Because really, you may get a relative on your mother's side reading your question and putting two-and-two together and then tell your mother that you're looking for your father. Is that what you want?
Now, if after all this, you still want to find paternal relatives on your own without help of a professional, I'd suggest you check out the site DNAAdoption.com and study it thoroughly. They have a lot of instruction and tools there that, after you gain some expertise in Autosomal DNA analysis, can help you analyze your DNA to discover relatives on your paternal side.
