1) Original sentence: 日本語ができると旅行がしやすくなって便利だと思いますよ。
If I write it as 日本語ができると旅行するがやすくなって便利だと思いますよ。
Is it grammatically correct and does it mean the same thing? If it is wrong, why is it wrong?
2) Also, another example, 昨日買物をしに行きました If I write it as 昨日買い物していきました is it also the same meaning and is it grammatically correct
If it is the same meaning, is there any reason to use the sentence structure with し instead of suru/-te form?
日本語ができると旅行するがやすくなって便利だと思いますよ。
No, unfortunately your sentence is ungrammatical. For one, が cannot directly take a verb like する. が takes a noun, so you have to use a nominalizer の. Or you can just remove する and use 旅行 as a noun. For one, やすい in modern Japanese does not mean "easy" alone (although it did in classical Japanese). やすい means "easy-to-X" only when combined with a masu-stem of a verb. A fixed version would be like:
昨日買い物していきました
No. The original sentence means "I went out for shopping", but your sentence makes little sense. (It could mean "I shopped and then went there" or "I shopped before going there" when there is enough context). The te-form does not mark a purpose like the に as in the verb + に + motion verb construction.
