People who say they oppose Calvinism are generally responding to the TULIP acrostic and the first-level expansion of the words underneath it: Total depravity, etc. But they don't really respond to the content. It's just like any Biblical expositor who uses aliteration or rhyming or some other literary device to make the points of his sermon memorable: His actual points are contained in the way his outline is filled out, not in the outline itself. Let's take them one at a time along with some of the common objections.
Total Depravity: If you object to Calvinism at all, you must, logically, start here, since this is the first of the two streams from which the rest all follow. What is Total Depravity? It means there is nothing in you that God admires or accepts, and there is no act you can perform (including receiving Christ) that is not mixed with sin and therefore unacceptable to God.
Unconditional Election: If everything we are and everything we do is mixed with sin, then God's acceptance of anyone who sins must be without respect to anything we do. Many people who say they reject total depravity accept unconditional election, as long as it's for everyone who willingly receives Christ. (I once had a pastor who asserted this.) If you don't see the contradiction in that statement, you aren't paying attention.
Limited Atonement: This is where the argument really heats up. Did Christ die for all sins of everyone for all time? Let's not waste time but get right to the question you will inevitably hear on this issue, a kind of "Christ available to all" trump card. I John 2:2 says, "He is the propitiation for our sins, and not for ours only but also for the sins of the whole world." This verse divides all the world into two camps, "ours" and "the whole world." To assert that Jesus died for all the sins of everyone everywhere for all time from this verse requires logically that you assert one of two things:
These are both unBiblical assertions. The answer lies in the use of the pronoun form, "ours". Do some homework and examine how John uses the second person plural pronoun forms throughout the first two chapters. Then come back and let's talk.
And to find out what else the Bible says on this subject, I suggest starting with the 6th, 10th and 17th chapters of John's Gospel.
Irresistable Grace: The usual objection here is something like, "God didn't make us robots who are unable to respond of our own free will." This is true, but it doesn't go far enough. Because of Adam's sin, we are unable to respond to the Gospel, see John 6:44 and 6:65 and Romans 5. That's why, in Ezekiel's picturesque language, God says, "And I will give you a new heart, and a new spirit I will put within you. And I will remove the heart of stone from your flesh and give you a heart of flesh."
By way of analogy, I have the freedom to flap my arms and fly, but I lack the ability. Or, closer to the subject, in the words of the old hymn, "'Do this and live,' the Law demands, but gives me neither feet nor hands. A better way His grace doth bring: It bids me fly and gives me wings."
"But how can you call that freedom?" you ask. In the words of C.S. Lewis, it's the way a man who is drinking is freed from thirst.
Perseverance Of The Saints: (Some folks call this "Preservation of the saints".) Many, including the pastor noted above who wanted unconditional election without total depravity, want to claim this one without its predecessor. But that falls apart, because if grace can be successfully resisted preventing salvation, it must also be resistable to the point of being able remove it.
Then what of those who leave their faith? I John 2:19 tells us, "They went out from us, but they were not of us; for if they had been of us, they would have continued with us. But they went out, that it might become plain that they all are not of us."
Okay, now that we're talking about the same thing, let's talk about your objections to Calvinism.