του αρέσει να ταξιδεύει
This sentence is in the Travel module. I would have expected it to be του αρέσει να ταξιδέψει and mean "he likes to travel" (i.e. go somewhere, stay there for a while and then come back). Then του αρέσει να ταξιδεύει would mean "he likes to be traveling" (i.e. go from one place to another without coming back). So the question is, am I correct, or is my understanding perhaps based on some grammar rules which native speakers don't really respect, or what?
The basis of your assumption, that "to travel" = να ταξιδέψει and "to be travelling" = να ταξιδεύει is correct.
However, this does not work when these follow the verb to like in the present tense. When you like to travel, you like travelling in general, so in the present tense 'like to travel' and 'like to be travelling' is the same. And this can only be translated as του αρέσει να ταξιδεύει, as in Greek there is only one present tense.
There would be a difference if we said "He would like to travel this summer". Then, yes, θα του άρεσε να ταξιδέψει. Then, it would also be "he would like to be travelling every summer" = "Θα του άρεσε να ταξιδεύει κάθε καλοκαίρι". In this case, it's about expressing a wish, so the whole grammar changes to reflect that. Consider the same example by replacing would like with want. Again, there is a difference if you're talking about what someone wants to do:
Θέλει να ταξιδεύει = He wants to be travelling in general (Actually I think even the simple infinitive works in English here - don't quote me on this!)
Θέλει να ταξιδέψει = He wants to travel (there is a point of reference in time implied here: e.g. this summer, before he's too old to do so etc.)
There is a lot of mismatching between the tenses in the two languages, especially in cases like this. I think the examples here in DL are a bit vague about the continuous aspect exactly because of that, e.g. https://www.duolingo.com/comment/17858591. Also, it's a bit difficult to explain it without setting rules that are too strict to lead anywhere and I sure hope what I've written above makes some sense! Any input and corrections are of course welcome, as I may have missed even something quite obvious in either language- it is late and I'm going to bed!! :D
The verb "ταξιδεύω" means broadly to travel, not necessarily to go somewhere and come back.
The phrase "του αρέσει" is going with a habit, with a repeated action. The sentence "του αρέσει να ταξιδέψει" is wrong because "ταξιδέψει" refers to a non repeated action. "ταξιδέψει" means that someone will travel for only one time.
If you want to say "he/she wants to travel for only one time" then you should say "θέλει να ταξιδέψει".