How to predict precipitate upon evaporation from calcium ions and either chloride, carbonate, or nitrate counterions?
This question is testing your knowledge of general solubility trends, which largely need to be memorized. Here is a fairly comprehensive list, and there are many others scattered through the internet.  

You are correct that nearly all nitrates are soluble, so that would be a bad guess.  

The halides are generally pretty soluble. There are more exceptions than for nitrates, but as a rule halide salts tend to be soluble.  

Carbonates however are frequently insoluble. The Group I cations tend to be an exception, but not the Group II cations, like $\ce{Ca^2+}$.

So, the general trend in solubility for the three anions in your question is $\ce{NO3- &gt; Cl- &gt; CO3^2-}$, so you would expect $\ce{CaCO3}$ to precipitate first.  

We can verify this against the known solubilities of each of the salts at room temperature:  

$\begin{array}{rr}
\pu{\ce{Ca(NO3)2}}: &amp; \pu{1212 g/L }\\
\pu{\ce{CaCl2}}: &amp; \pu{75g/L}\\
\pu{\ce{CaCO3}}: &amp; \pu{0.01g/L}
\end{array}$ 

So, the quantitative solubility data confirms the original prediction and $\ce{CaCO3}$ will clearly precipitate first.