A confusing topic! The most commonly used romanzation system in both Japan and the West (the Hepburn system) accurately transliterates the sound "shi" as "shi." However, in the Japanese-developed but less commonly used transliteration systems (Nihon-shiki and kunrei-shiki) there's a closer correspondence with Japanese writing (kana) systems, but the way a Westerner would read it does not always follow the sounds correctly. For example, in Nihon-shiki the sound "shi" is written "si," and the sound "tsu" is written "tu." I always think Nihon-shiki looks funny.
Takahashi (meaning "high bridge") is a common family/place name in Japanese, and is usually transliterated with "shi." That's the way it would appear in a Japanese passport or on a road sign. So, I strongly reccomend "Takahashi"