The Biblical Case
Genesis 1 clearly describes 6 consecutive days of creation delineated with the "evening and morning" phrase for each one.
Genesis 2 starts with declaring that the creation of the heavens and earth was complete and then God rested the 7th day.
The rest of Genesis 2 is about the account of the creation of humans specifically and combined with a little translation ambiguity it sometimes gets declared as a contradiction with Genesis 1 in the order in which things happened. Here is an article giving a full explanation.
The next significant point in the Bible that helps us examine the validity of the creation account is in the 10 Commandments. Exodus 20:11. This is the words of God himself telling us specifically that the heavens and earth were created in 6 days and to remember the Sabbath day as it is a remembrance of the creation and of our creator. What is especially critical to understand here is that the Sabbath set forth here is every week as we know it. The Jews(and some other groups) still hold to the Sabbath day in the manner it is presented in the Bible. So it is very clear that God is specifically saying that His creation was just that one week.
I will skip ahead to the words of Jesus now. Matthew 5:17-19. Jesus states that not one bit of the law will change or be done away with till heaven and earth pass away and all is accomplished. Well, if that is the case then the 10 Commandments were totally validated by Jesus himself and, again, Exodus 20:11 is part of the 10 Commandments and clearly states creation took 6 days as we know them as he reminds the Israelites of the Sabbath.
Jesus again says something in Matthew 19:4,8 that is relevant to the conversation. He states that humans were created at the beginning. As Genesis states, humans were created on the 6th day of the creation week and that would certainly qualify as "the beginning." What wouldn't qualify, certainly, is the evolutionary path where humans come along very late in the creation sequence.
Jesus states something else that continues to affirm the biblical creation account. In John 5:46-47 Jesus questions how one could believe in Him if not Moses. "Moses" being the first 5 books of the Bible attributed to being authored by Moses. This, of course, includes Genesis and Exodus.
So if you are a Christian and believe in Jesus as God who came to earth as a sacrifice for our sins and to establish the church here on earth and show us the way to Him and salvation, well, he himself says this part of the Bible is true.
Further, Jesus makes other statements about his validity. Matthew 7:24-26 Jesus states that following his words is wise and ignoring them is foolish. Also these: John 3:16, John 6:68-69, John 8:12, John 8:31, John 10:27, John 15:5-6, John 15:14.
2 Peter 3:8 is sometimes mentioned as a statement that the days mentioned in Genesis don't actually mean 24 hour days as we know them. Couple problems with this idea. First is the context of 2 Peter 3:8. 2 Peter 3:3-7 is pretty strong language in support of the YEC viewpoint. Going so far as to mention uniformitarianism and calling out those who push it as "scoffers." So Peter makes this strong statement in support of the biblical viewpoint then immediately turns around and says something that doesn't fit? No, what he is really saying is that we must be patient and that just because its been a while since creation, and from our perspective now since Jesus was here on earth, that we shouldn't lose hope in 2nd coming. He isn't stating a conversion ratio of days to years or whatever. Another problem is that even if this statement is intended as a conversion ratio, the amount of time this is surmising is still dramatically short of the supposed ages by the mainstream viewpoint. 6 thousand years is nowhere near the millions of years the mainstream view pushes.