1. 'I travelled to...' tells us about something that you did at a specified point in the past. A conversation might go like this:
What did you do last summer?
I travelled to Mexico.
'Last summer' is a finished or 'closed' time period, so we use the past simple. You cannot use a present perfect (have travelled) with a finished time period.
2. 'I have travelled to' is not about any specific point in time. The present perfect is related to the present moment in some way, and refers to an 'open' or unfinished time period. We often use the present perfect to talk about experiences in your life so far. For example 'I've done lots of interesting things in my life. I've travelled alone across South America, I've met many fascinating people ..'
3. 'I've been..' refers to the same time period as 'I have travelled', but the emphasis is on being in a particular place rather than on travelling there. Note that we tend to use 'to', rather than 'in' with this phrase, as in 'Have you ever been to Mexico?'. Again, this is about your general experiences in your life up to now, rather than an event at a specific point. Here's a typical conversation:
Have you ever been to Mexico?
No I haven't. Have you ever been there?
Yes, I have, in fact. I travelled across South America when I was a student, and I spent a few weeks in Mexico. It was amazing.
As you can see, the conversation starts in the present perfect when we are talking about general life experiences, and then switches to the past simple when a specific time in the past is mentioned (when I was a student).
I hope that helps.