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Note: this is a homework problem so please don't give me the whole answer!

I have two variables, A and B, with normal distributions (means and variances are known). Suppose C is defined as A with 50% chance and B with 50% chance. How would I go about proving whether C is also normally distributed, and if so, what its mean and variance are?

I'm not sure how to combine the PDFs of A and B this way, but ideally if someone can point me in the right direction, my plan of attack is to derive the PDF of C and show whether it is or isn't a variation of the normal PDF.

Hopefully it's clear to you that C isn't guaranteed to be normal. However, part of your question was how to write down its PDF.
@BallpointBen gave you a hint. If that's not enough, here are some more spoilers...

Note that C can be written as:
$$C = T cdot A + (1-T) cdot B$$
for a Bernoulli random $T$ with $P(T=0)=P(T=1)=1/2$ with $T$ independent of $(A,B)$. This is more or less the standard mathematical translation of the English statement "C is A with 50% chance and B with 50% chance".

Now, determining the PDF of C directly from this seems hard, but you can make progress by writing down the distribution function $F_C$ of C. You can partition the event $C leq X$ into two subevents (depending on the value of $T$) to write:

$$ F_C(x) = P(C leq x) =
P(T = 0 text and C leq x) + P(T = 1text and C leq x) $$

and note that by the definition of C and the independence of T and B, you have:

$$P(T=0text and C leq x) = P(T=0text and Bleq x) = frac12P(Bleq x) = frac12 F_B(x)$$

You should be able to use a similar result in the $T=1$ case to write $F_C$ in terms of $F_A$ and $F_B$. To get the PDF of C, just differentiate $F_C$ with respect to x.

Simulation of a random 50-50 mixture of $mathsfNorm(mu=90, sigma=2)$ and
$mathsfNorm(mu=100, sigma=2)$ is illustrated below. Simulation in R.

set.seed(827); m = 10^6
x1 = rnorm(m, 100, 2); x2 = rnorm(m, 90, 2)
p = rbinom(m, 1, .5)
x = x1; x[p==1] = x2[p==1]
hist(x, prob=T, col="skyblue2", main="Random 50-50 Mixture of NORM(90,2) and NORM(100,2)")
 curve(.5*(dnorm(x, 100, 2) + dnorm(x, 90, 2)), add=T, col="red", lwd=2)

One way you could work on that is to analyze it as the variance tends to 0. This way you would get a Bernoulli-like distribution, which is (clearly) not a normal distribution.

If the PDFs of $A$, $B$ and $C$ are $P_A(x)$, $P_B(x)$ and $P_C(x)$ respectivelly, and $alpha$ is the chance that $C$ is defined as $A$ (and $1-alpha$ is the chance that $C$ is defined as $B$), then

$$P_C(x) = alpha P_A(x) + (1-alpha) P_B(x).$$

This is the kind of problem where it is very helpful to use the concept of the CDF, the cumulative probability distribution function, of random variables, that totally unnecessary concept that professors drag in just to confuse students who are happy to just use pdfs.

By definition, the value of the CDF $F_X(alpha)$ of a random variable $X$ equals the probability that $X$ is no larger than the real number $alpha$, that is,
$$F_X(alpha) = PX leq alpha, ~-infty < alpha < infty.$$
Now, the law of total probability tells us that if $X$ is equally likely to be the same as a random variable $A$ or a random variable $B$, then
$$PX leq alpha = frac 12 PA leq alpha + frac 12 PB leq alpha,$$
or, in other words,
$$F_X(alpha} = frac 12 F_A(alpha} + frac 12 F_B(alpha}.$$
Remembering how your professor boringly nattered on and on about how for continuous random variables the pdf is the derivative of the CDF, we get that
$$f_X(alpha} = frac 12 f_A(alpha} + frac 12 f_B(alpha} tag1$$
which answers one of your questions. For the special case of normal random variables $A$ and $B$, can you figure out whether $(1)$ gives a normal density for $X$ or not? If you are familiar with notions such as
$$E[X] = int_-infty^infty alpha f_X(alpha} , mathrm dalpha,
tag2$$
can you figure out, by substituting the right side of $(1)$ for the $f_X(alpha)$ in $(2)$ and thinking about the expression, what $E[X]$ is in terms of $E[A]$ and $E[B]$?