The cosmological argument argues for the existence of an uncaused cause that is eternal and unchangeable. There are two main responses to this argument.
1) That doesn't mean that the uncaused cause is Jesus.
There's no particular reason to argue with this response. We know about Jesus from the Bible, not from the cosmological argument.
2) If God exists as an uncaused cause, why not just assume that the universe itself is an uncaused cause?
The answer here is a bit nuanced. The cosmological argument shows that nature does not admit of an uncaused cause. Therefore, we must turn to the supernatural. However, just because the supernatural must have brought nature into being does not mean that nature can bring itself into being. In other words, this second response is an example of category error.
Let me try to clarify that a bit further. One of the foundation principles of the cosmological argument is that nothing comes from nothing. Yet, we see and say that God made all things from nothing. Does this disprove one of the premises upon which the cosmological argument is founded? No - because "nothing comes from nothing" is the way that nature operates. God is supernatural. Thus, the fact that God brings something from nothing does not violate the principle.
Perhaps it becomes even easier when you look at it this way. The cosmological argument is an argument for the fact that a creation miracle must have occurred. However, that does not mean that miracles are an ordinary part of nature. If you see a turtle on a fencepost, you don't think, "wow, I guess turtles CAN climb fences," you think, "someone put the turtle there."
You can easily see that it is absurd to say: "if someone can put a turtle there, why not just assume that the turtle can put himself there?" Yet, it is less obvious (particularly to atheists) to say that "if God can create the world from nothing, or be self-existent, or be timelessly eternal - then why not assume that the universe can be the same?"
I'm sure there are plenty of other responses, but if you ever find yourself in a discussion in which the cosmological argument comes up, perhaps you'll be able to help folks see that the universe is like a turtle on top of the fence post. Someone must have put him there - we know that because the turtle can't possibly have put himself there - just as we know that something outside of nature must have caused nature to exist.
And that something - whose identity is outside the argument - is the Lord God.