Question: Which one is correct? Can I use both?

"I am going to listen and read."  

"I am going to listen and to read."

Answer: 
Depends on how distinct you want *listen* and *read* to sound. Tying both objects to one infinitive conveys the sense of a single activity, while preceding each object with *to* conveys distinct activities.
It really depends on how you group the *to* and what you intend it to mean.

If you are using the phrasal verb "going to" as in:

> 
> What are you going to do when you get there?  
> 
>  I am [going to] listen and read.
> 
> 
> 

But if you are just using the verb "go" as in:

> 
> Why are you going to the exhibition?  
> 
>  I am going [to listen] and [to read].
> 
> 
>