What's the correct way: "Helped me go through" or "Helped me going through"?

Solution 1:

The first one is more idiomatically common, and I believe that is indeed what you mean to say. I will analyse the difference between the two.

(In the following, we will ignore the alternative wording suggested by the contributors in the comment section -- they may or may not be improvements, but that's irrelevant to the immediate question you are asking.)

The first one literally means: "Good friendships helped me to go through tough times."

That is, the help is what is needed in order to be able to go through the tough times.

The second one literally means: "Good friendships helped me while I was going through tough times."

That is, you would have gone through the tough times whether or not your good friendships were around to help you, but they did make it easier for you.

The difference is subtle, and I believe that you are correct in your understanding that the first one is what you meant.