Now that I have been studying at DMM, I feel like travelling to different countries.
Since I started studying English at DMM, I have become much more interested in visiting lots of different countries.
ーNow that I have been studying at DMM, I feel like travelling to different countries.
「DMMで勉強してきた今、いろんな国に旅行したい気分です」
to travel to different countries で「いろんな国に旅行する」
ーSince I started studying English at DMM, I have become much more interested in visiting lots of different countries.
「DMMで英語を勉強し始めて以来、たくさんの国を訪れたいと思うようになった」
to become much more interested in ... で「…にもっと興味を持つようになる・もっと〜したくなる」
ご参考まで！
Since I started DMM, I’ve wanted to visit many different countries.
Since I started DMM, I’ve wanted to visit many different countries.
これは
「DMMを始めてから、いろいろな国に行きたくなりました」
という意味です。
visit many different countries は
「いろいろな国を訪れる」
という自然な言い方です。
例文
Since I started DMM, I’ve wanted to visit many different countries.
DMMを始めてから、いろいろな国に行きたくなりました。
Since I started studying English, I’ve wanted to travel more.
英語の勉強を始めてから、もっと旅行したくなりました。
カナダ 
アメリカ合衆国 
