Origin of the Earth
Where you might hear about it: Earth Science/Geology, Astronomy, even a biology course
What they'll tell you: the Big Bang story, the Nebular Hypothesis, how earth started out as a molten ball of magma, and that life started out as a single cell in a primordial soup. This life eventually played a major role in changing earth's features and atmosphere and making it habitable for modern life forms.
What to remember:
i. The Big Bang is logically impossible. You cannot get something from nothing.
ii. Yes, God could have used the Big Bang, but why tell us the whole creation week account thing if He did? And after all, Jesus did say people were around "at the beginning of creation" (Mark 10:6) not "long, long after the beginning of creation."
iii. The Nebular Hypothesis has some problems as well, not the least of which is that most other solar systems have gaseous planets in the inner orbits and small, rocky planets in the outer orbits, and the same theory can't really explain different distributions of densities and compositions. The Nebular Hypothesis says that the planets coalesced from spinning debris around the sun. That should result in consistent patterns of planets: big ones near the sun, small ones out (or vice versa); dense ones near the sun, lighter ones farther out (or vice versa) - in short, there should be one pattern of planets being ordered if they all formed from the Nebular Hypothesis. The diversity of planet systems speaks against it.
iv. The origin of life from non-life is something called abiogenesis. It's biologically impossible and has never been observed in a laboratory. The Miller-Urey experiment they love to tell you about did not create life (it created a few chemicals good for life and a few toxic to it) and required conditions that are nothing like what the early earth's environment were supposed to be like.
Where you might hear about it: Biology, Earth Science/Geology, Anthropology, Psychology, and maybe a little in Philosophy
What they'll tell you: all living things evolved from one original cell. Natural selection and mutations built up over millions of years to diversify into algae, fungi, simple animal phyla and later, the land plants and advanced chordate animals. The fossil record proves evolution by showing the gradual transition from simple organisms into more advanced ones over the past 3.8 billion years.
What to remember:
i. Natural selection and mutations do occur. Speciation occurs. The Galapagos Island Finches really did diversify into various species from one original finch type. These examples of observational science, however, do not mean that bacteria could evolve into baleen whales. They mean that within any particular type, a great deal of diversity and speciation can occur.
ii. Natural selection selects from information that is already available in the genome. It can emphasize traits, it can weed bad traits out of the gene pool, but it cannot add new information.
iii. Mutations are responsible for genetic anomalies, inherited diseases, and the general decay of living things over time. Cave fish becoming blind by mutations may be beneficial in the moment, but overall the fish decayed over time because they lost their incredibly complicated eye structures. So mutations can lead to features that are beneficial, but only beneficial in extremely specialized environments. Overall, and outside of those specialties, they are negative.
iv. Many living systems and structures are what we call irreducibly complex. For example, the biochemical systems and reactions involved in mitochondria, DNA replication, blood clotting, and other workings in and among cells. There are so many parts to make these systems work, that you can't take a single piece away without ruining the whole system. Such structures cannot evolve gradually, step-by-step, over millions of years.
v. The Theory of Evolution cannot explain the origins of sex. Not only do you need two complex and distinct reproductive systems, but you need these two wholly different systems to show up - at the same time - randomly - and somehow be exact and perfect complements of each other. Sexual reproduction is a pretty major driving force of change and diversity in evolutionary theory (we could not have the diversity and robusticity of species today if there was only asexual reproduction), so the fact that evolutionary theory cannot explain this is pretty much a death blow.
vi. The fossil record is full of missing links. The few "transitional" fossils found either have one or two special features whose importance has been blown out of proportion (like A. sediba), or they have been unique but totally non-transitional (like the platypus or coelacanth), or they have been outright fraudulent (Piltdown Man).
vii. Fossils in the fossil record do not come with dates attached to them. You have many fossils out-of-sequence (birds preexisting Archaeopteryx), and many of the dating methods used on the fossils, and the rocks they are found in, contradict each other and thus cannot be relied upon.
viii. Change occurs within types of living things (types, aka baramins or families, include: Dogs & Wolves. The rose family. Finches. Etc.). However, types cannot evolve into other types (dinosaurs into birds, or wolves into whales, or seaweeds into ferns).
The Age of the Earth
Where you might hear about it: Earth Science/Geology, Astronomy, Biology, and just about any other science class or other class with material even remotely science-related.
What they'll tell you: The earth is 4.8 billion years old. Astronomy proves this. Geology proves this. The evidence for several successive ice ages proves this. The fact that all living things evolved from one cell proves this.
What to remember:
i. The sun is gradually getting brighter. That doesn't really affect us, and it hasn't for all of human history. 3.8 billion years ago, however, when life was just getting started, it would have been so much dimmer that life would have had a very hard time getting started. We call this the Faint Sun Paradox.
ii. The moon is gradually spiraling away from us. It is supposed to be the same age as the earth, but in 4.8 billion years it should be much farther away from us than it is now.
iii. Radiometric dating can be done with multiple radioactive isotopes. Experiments have shown that using different isotopes to date the same rock can lead to such vastly different ages as a couple million years old to a couple billion years old. Such wild contradictions must mean that radiometric dating has some problems and cannot be used to "prove" an old earth.
iv. C-14 dating can only date things up to 100,000 years old (it's gone after that many years), but it has been found in objects - fossils, coals, diamonds - that are supposed to be hundreds of millions or even billions of years old.
v. Fossils of trees were C-14 dated to be a couple thousand years old, but other radioisotopes were used to date the rocks the tree fossils were found in. The rocks were dated as millions of years old. Again, this is a serious contradiction epitomizing the problems with radiometric dating.
vi. Geologic structures like canyons, rock layers, etc. have been observed to form at a rapid pace in geologic catastrophes. This means that you don't necessarily need millions of years to get the Grand Canyon or Arches National Park. Geological formations do not require an old earth.
vii. Instead of there being several ice ages, there was probably only one. Secular theories cannot explain how an ice age could even get started (how do you evaporate a ton of water and then cool it into snow? You need hot temperatures AND cool temperatures simultaneously, and that's really hard to explain) but creation theories explain it nicely. There was one ice age after the Great Flood and you do not need millions of years to get one.
viii. Evolution requires an old earth because it is easier to believe that pond scum and turn into a human being if you give the pond scum a couple billion years to accomplish such a feat. Without an old earth, the Theory of Evolution is ridiculous, and that is why scientists grasp at straws (and contradictory dating methods) to believe in one.
Where you might hear about it: Anthropology, Psychology, Biology
What they'll tell you: Humans diverged from the Great Apes from a common ancestor about two million years ago. Transitional forms like Lucy and the Neanderthals show the transition from apes to bipedal, walking ape-like hominids to larger-brained hominids to people. Humans today come from a group of such hominids in Africa (or possibly Asia).
What to remember:
i. Lucy is an ape. Her hips are comparable to apes who walk upright some of the time, not all of the time. She was not bipedal.
ii. Neanderthals and the Denisovans were people. They buried their dead (complete with scattered flowers), they made tools, they performed dental operations on each other, they even made mathematically precise flute-like musical instruments. They were human in every possible way. And, if they were around today, they'd be slamming the racists who call them ape-men.
iii. The notion that chimps and humans share 98\% of their DNA is a notion based on the outdated and refuted idea of "junk DNA." If you compare the total human genome to the total chimp genome - without ignoring the "junk DNA" that was previously ignored in each - you will find that it's closer to 80\% or even less of a shared genome. Furthermore, a shared genome is not proof of a common ancestor. It's proof that living things share much of the same biochemistry (we all have mitochondria, and DNA, and proteins, and we read and express much of DNA the same way*). Similarities in DNA prove that living things live off of a common language, and that's as much evidence of a common designer as it is for a common ancestor (if not more so).
For More Information
Evolution Exposed: Biology and Evolution Exposed: Earth Science, both by Roger Patterson, are excellent resources that expose the holes in evolutionary, origin of the universe, and age of the earth theories. While aimed at high school students (the books quote liberally from high school science textbooks), they go in more than enough detail to be useful field guides for worldviews and inconsistencies in your college classes.
*So, cool fact. You know those glow-in-the-dark GloFish you see in pet stores? They glow because scientists took genes for bioluminescence out of jellyfish genomes and spliced them into the fishes' genomes. Jellies (Cnidarians) and fish (Chordates) are supposed to be separated by over 500 million years of evolution, and yet they read this gene and express this gene in the exact same way.
How is that even possible? ;)