In my admittedly limited experience, you will be very lucky if you can actually identify relatives who share the same surname as you, your parents, grandparents or even their immediate ancestors, more often that not. It's more a matter of trusting that over a certain match threshold, your chances of being related are significant to a very high percentage, but you have to put some work in to find the relationship.
You can start by taking your tree back as far as you can go and then working forward again on as many collateral lines as you can, to as near the present time as possible, as 3rd, 4th and 5th cousins may well have names you have never heard of, but can still be instrumental in confirming the accuracy of your other research.
The amount of DNA that you share with a possible match can be entered on the DNA Painter site, and will help with an indication of the possible relationships you are likely looking at, and how many generations back you need to consider exploring. The tools available on GEDmatch are also very useful, particularly after you have obtained a few proven matches and can compare chromosomes and segments that you match on, against those for new matches, that can indicate the possible family line that connects you.
As an example, I explored a match with someone whose name meant nothing to me. We exchanged parents and grandparents names, and none of those meant anything to either of us, but she had several relatives who came from the same area of the country as my grandmother. I took that line of her family further back using GRO and census searches, but it petered out when it went over to Ireland, and I have no known links in that direction.
Then I happened to notice that one particular ancestor of her mothers, who I will refer to as Mr. SMITH, three generations back, had been born in a town where I knew that the family of the person who I believed to be my orphaned grandfather were from. My matches' tree hadn't been explored beyond that point, and as I wasn't that sure about the identities of my grandfather's parents and ancestors, that part of my tree was pretty sparse as well. So I set about researching those lines in both my tree and my possible matches' tree. I found that a Miss SMITH had married one of my suspected Gx3 grandfathers, and that one of Miss SMITH's brothers was a direct ancestor of my matches' grandmother.
So I had a set of shared Gx4 grandparents with that match, which supported the link to my suspected grandmother. I also investigated another possible match with a name that again meant nothing to me, and he similarly turned out to be a 4th cousin linking me to my grandfather's suspected paternal line. Neither match on their own would have been wholly conclusive, but with matches linking me to both maternal and paternal sides of the family, I'm now pretty happy that my suspicions are correct. If I'd just looked at the names and discounted both matches on the basis that they meant nothing to me, I'd still have no idea where my grandfather was from!
I can also recommend a book called Tracing Your Ancestors Using DNA, edited by Graham S. Holton. It covers DNA investigation in some depth, but in a way that makes it easy for someone with no previous knowledge to understand, and I found it well worth reading and still reference it from time to time.
So my advice would be not to get dispirited, but don't expect matches to simply fall into your lap and have everything fit straight into place. You may have to do some (or quite a lot) of investigative work, either jointly with your match or perhaps investigating their tree yourself, before you find the links.
Good luck!
