「[日本]{に・ほん}にお[寿司]{す・し}を[食]{た}べに[行]{い}きたい。」(1)-A
In this case, the speaker's purpose of visiting Japan is eating sushi. This sentence clearly expresses that the speaker's motivation to go to Japan is sushi.
「日本に行って、お寿司を食べたい。」(2)-A
can have the same nuance as (1)-A above, but it also can imply eating sushi is one of the purposes of visiting Japan. it depends on the tone and the context.
「日本でお寿司を食べていきたい。」(3)-A
Using に instead of で in this sentence is unusual, so I explain the nuance of で-version of the sentence. (By the way, what user4092さん mentioned in comment is true. 日本にお寿司を食べていきたい literally means so. This composition of sentence is used sometimes, for example, when someone says [映画]{えい・が}に[夕飯]{ゆう・はん}を食べていきたい which means 映画に夕飯を食べてから行きたい(I want to go to movies after have dinner). If someone means so, the composition of 「夕飯を食べて、映画に行きたい」 or 「お寿司を食べて、日本に行きたい」 is preferred, though.)
When someone says this way (「日本でお寿司を食べていきたい。」), there are other plans in her/his mind. The other plans might be stopping by other places, or visiting the next place, or how to go back to her/his country, or something like that. The speaker has something else to do in the trip, but anyway, in Japan, s/he wants to eat sushi. Eating sushi in Japan is not the whole plan the speaker has. It's a part of the itinerary the speaker plans. So this sentence implies that the speaker visits Japan from other country and will leave Japan after s/he eats sushi because s/he has other plans. In other words, it expresses the nuance of "go, do(eat sushi), and then leave for the next place."
The sentence (3)-A could also mean "In Japan, at least I want to eat sushi." Whether the "at least" nuance is expressed depends on the tone and the context.
を VS は
The "at least" nuance becomes obvious when は is used instead of を.
「日本でお寿司は食べていきたい。」(3)-B
This sentence implies that eating sushi has the top priority in the speaker's to-do list of trip to Japan.
The sentence (2)-A could not express the "at least" or " top priority" nuance. It simply describes that eating sushi is what the speaker wants to do. But if は is used instead of を such as (2)-B, it could.
「日本に行って、お寿司は食べたい。」(2)-B
If は is used instead of を in the sentence (1)-A, such as (1)-B, it implies that the speaker wants to go to Japan in order to eat sushi but there are other things s/he doesn't want to do in Japan.
「日本にお寿司は食べに行きたい。」(1)-B
The (2)-B and (3)-B sentences also could imply the same nuance as (1)-B. It depends on the tone and the context.
Omission of を and に
In colloquial Japanese, を and に are often omitted when they are directly followed by their verbs (there are some exceptions though). For example, in the sentence (2)-A, 日本に's verb is 行く which directly follows 日本に, and お寿司を's verb is 食べる which directly follows お寿司を. So these を and に can be omitted like this.
「日本行って、お寿司食べたい。」(2)-C
This expression is natural and common in casual conversations.
On the other hand, in the sentence (1)-A, 日本に's verb 行く does not directly follow 日本に. There is a phrase お寿司を食べに between 日本に and 行きたい. So this に in the 日本に is never omitted, while the を of お寿司を食べに is omittable as well as (2)-A's を above. So it can be said
「日本にお寿司食べに行きたい。」(1)-C
Some people, usually young people, even omit the に of 食べに like this.
「日本にお寿司食べ行きたい。」(1)-D
But this type of に omission is very casual and not something most people do. The omission of に in verb-に-verb sequences is much less common than those in noun-に-verb sequences.
The (3)-A's を also can be omitted like this.
「日本でお寿司食べていきたい。」(3)-C
So, in real conversations, you may encounter these omitted versions of sentence. But, although they don't contain を and に, they have the same meanings and nuances as the original sentences which contain を and に.
