I'm going to travel to a country that I went last year.

[Mj_Developer]

Is the following sentence correct?

I'm going to travel to a country that I went last year.

[Beeesneees]

You need another 'to'
I'm going to travel to a country that I went to last year.

If you wish to avoid the preposition repetition, you could replace 'went to' with 'visited' or replace 'travel to' with 'visit'.
I'm going to travel to a country that I visited last year.
I'm going to visit a country that I went to last year.
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Cheers m' dears!

[Mj_Developer]

And what about the following sentence. Is it correct?

Next month, I'm going to travel to a country that I visited last year.

[Beeesneees]

That's fine. All you've done is add a specific future timeframe to the sentence I posted. The addition doesn't affect the tense there.
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Cheers m' dears!