What statistical test for my data?

I have a data set with three variables that are about different measures of the studied phenomenon (counts of words and other components in a sentence): var1 and var 2 (dependent variable) are continuous with float variables and var3 (independent variable) is continuous with integers (discrete counts). What is the right statistical test for my data to see how var1 and var3 as well as var2 and var3 are correlated? I thought about One way Anova. Would that be appropriate? To give you an idea about my data here is a small sample:

var1 var2 var3
0.25 141.421 1
0 200 0
0 141.421 0
0.1 223.607 0
0 173.205 0
0.5 100 1
0.083 346.41 2
0.154 360.555 2
0.214 264.575 0
0 200 0
0.143 264.575 0
0.062 282.843 1

Thanks in advance for your help.
Thank you for your question. What I am trying to show is that var3 (count of relatives) increases with the increase of occurrences of specific types of coordination words var1 or var2 (independently from each other). I tried Pearson test of correlation and the results were positive but I am not sure this is the right test.

Your help is appreciated.
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New Member
Just a followup question. What effect size measure should I use with the Poisson Regression? I would like to compare (or rank) my independent variables. So, I am wondering if Exp(b) can help determine which significant independent variable has a bigger effect on the change of my dependent variable. Other effect size candidates I've seen online are Cohen d, but I am not sure if it is a good choice for my data described above.

Thanks in advance for your help.
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