Well, why not turn my comment on Abhinav's answer into an answer?
$\newcommand{\ra}{\rightarrow}$
$\newcommand{\PP}{\mathbb{P}}$
$\newcommand{\F}{\mathbb{F}}$

This is a cut-and-paste from a passage in a paper I wrote earlier today.

Lemma:
Let $L/K$ be a purely transcendental field extension.

a) Let $V_{/K}$ be an algebraic variety. Suppose *either* that $K$ is infinite or $V$ is complete. Then $V(L) \neq \varnothing \implies V(K) \ \neq \varnothing$.

b) For every abelian variety $A_{/K}$, we have $H^1(L/K,A) = 0$.

Proof: a)
Step 1: Let $\{t_i\}_{i \in I}$ be a transcendence basis for $L/K$. If $P \in V(L)$, there is a finite subset $J \subset I$ such that $P \in V( K(\{t_i\}_{i \in J})$. Thus we are
reduced to the case in which $L/K$ has finite transcendence degree. Induction reduces us to the case $L = K(t)$.

Step 2: A point $P \in V(K(t))$ corresponds to a rational map $\varphi: \mathbb{P}^1 \ra V$. The locus on which $\varphi$ is not defined is a finite set of closed points of $\mathbb{P}^1$. If $K$ is infinite, so is $\mathbb{P}^1(K)$, and thus there is $P \in \mathbb{P}^1(K)$ at which $\varphi$
is defined, and then $\varphi(P) \in V(K)$. On the other hand, any rational map from a regular curve to a complete variety is a morphism, so if $V$ is complete then
e.g. $\varphi(0) \in V(K)$.

b) Since $\eta \in H^1(K,A)$ corresponds to a torsor $V$ under $A$ and thus a
projective variety, this follows immediately from part a).

Remark:
a) If in the statement of the Lemmma we strengthen "complete" to "projective", a more elementary proof can be given: let $\varphi: V \ra \mathbb{P}^N$ be a $K$-embedding. Since $K(t)$ is the fraction field of
the UFD $K[t]$, if $P \in V(L)$, we can write $\varphi(P) = [f_0(t):\ldots:f_N(t)]$ with $\operatorname{gcd}(f_0,\ldots,f_N) = 1$. In particular, some $f_i(t)$ is not divisible by $t$ and thus $(f_0(0):\ldots:f_N(0)) \in V(K)$.

b) Let $K = \F_q$ be a finite field. Then the affine curve $V = \PP^1_{\F_q} \setminus \PP^1(\F_q)$ has $K(t)$-rational points but no $K$-rational points.

There follows (in the manuscript) a result that if you start with $V_{/K}$ and replace $K$ by the function field $K(X)$ of a variety $X_{/K}$ with a rational zero-cycle of degree $1$, the least degree of a rational zero-cycle on $V$ does not change. Even this result is rather well-known, I think, but it can be hard to find these types of things written down in "proper generality".

Also the OP mentions something about his genus zero curve $C_{/K}$ having enough rational points to be rational. For this: it is not hard to show that a geometrically integral curve of arithmetic genus $0$ over an arbitrary field is birational to $\mathbb{P}^1$ if and only if it has at least one nonsingular $K$-rational point. By Riemann-Roch the curve is birational to a conic, and having a nonsingular rational point is a birational invariant (think in terms of valuations on the function field).