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Which sentences below are correct? Please explain why it is correct and why it is wrong.
These are the places where I want to travel.
These are the places where I want to travel to.
These are the places which I want to travel to.
Must "the" be used in all the sentences above?
There appear to be a couple of different things you're asking:
"travel" can just mean generally travelling around, without necessarily any particular destination, or possibly going to multiple places within some larger area, whereas "travel to (somewhere)" means you have one specific destination.
In the case of your examples, "where I want to travel" tends to imply that you want to travel around to multiple places within that country, while "travel to" just says you want to travel and (somewhere in) that country is the destination you are going to go to.
You can use either "which" or "where" in this case. In general, "which" means that the noun is actually the direct object of the following clause, but "where" is not as restrictive (it just means that it is the location associated in some way with the activity), so the two do not always mean exactly the same thing, and there are some cases when you can use "where" but you can't use "which" and vice-versa, but in this case they are pretty much equivalent.
Here's where we run into a bit of a problem, because it's generally not correct when you end up with a preposition at the end of a sentence. When you are creating a dependent clause using "which", and the clause involves the use of a preposition like "to", the correct way to do this is to actually move the "to" in front of the connecting preposition ("to which I want to travel").
However, attaching the preposition like this only works with "which" (not with "where", or with "that", etc), so if you're going to be using a clause with a preposition correctly, you need to use "which". "where" won't work in this case (so "where I want to travel to" is wrong, but you also can't say "to where I want to travel" either).
Having said that, many people do use dangling prepositions on the end of sentences all the time, even though it's not grammatically correct. This really only works in casual conversation, however, and will generally be considered wrong if you're doing any kind of more formal writing.
All of your examples are reasonably ok for casual conversation. The last two are technically grammatically wrong because of the dangling "to" at the end. If you want to be fully grammatically correct, you can say either:
These are the places where I want to travel.
or
These are the places to which I want to travel.
(There is no way to rearrange the second example to be grammatically correct, because technically "to" and "where" don't go together at all, but it still sounds reasonably fine for casual (technically incorrect) use.)
In these examples, the definite article makes what you refer to a closed group, as it were.
"These are places I want to travel to" does not necessarily mean you are giving a complete, exhaustible list.
"There are the places I want to travel to" suggests that it is a complete list of places.
"Travel to" means that the place you name is your destination.
The word "travel" alone in this context can mean the same as "travel in" or "travel around". "Travelling" is often used to refer to an extended period of vacation where you may go from one place to another.
