Matthew seems to have been written first.
The two classic listings of what ended up being the New Testament both have Matthew first. The first is the same as ours, while the second begins Matthew, John, Mark, and Luke. It makes sense Matthew is listed first for a reason, which may make more sense as you study the evidences.
The first 5 verses of Luke say "Inasmuch as MANY have undertaken to compile an account of the things accomplished among us, just as they were HANDED DOWN to us by those who FROM THE BEGINNING were eyewitnesses and servants of the word, it seemed fitting for me as well, having investigated everything carefully from the beginning, to write it out for you in consecutive order, most excellent Theophilus; so that you may know the exact truth about the things you have been taught."
So, there are already MANY compiled accounts already in existance; not one, not a couple, but many! So, Luke is out of the running for being first for sure.
But Polycarp (AD 70-155) quotes Matthew well. It's not clear at all that he definately quotes Mark or Luke. From Ignatius (AD 35-107) we have strong Matthean qoutes, while the ones from Luke and Mark are barely noticable, and, could hve actually been used to help create Luke.
The arguments FOR Markan priority are not evidential. There are NO scraps of Mark know in existance that predate about AD 350. There is a scrap of Matthew from AD 160, and a scrap of John from AD 120. As a side note, the copy of The Didache is dated to be AD 100!
But back to Mark, all the most popular arguments are conjecture at best, against an overwelming amount of qoutes from early church fathers, and dated scraps. Under good scrutiny, Mark is a reduction of Matthew, and a paraphrasing into common street language of Greek speaking people. I think its safe to say Matthew was written first, and then John, followed pretty closely by Luke & Mark.