Assuming you have not misheard it, grammatically, some verb is omitted after を because を is the object marker. So it says "I _____ the beginning with you, forever."
Without a bit more context, this catchphrase does not make much sense. But this will make perfect sense as the catchphrase of One Week Friends, for example, where the memory of the main character lasts only for a week and she has to experience the beginning (of a romantic relationship) again and again. In this case, the missing verb would be something like 体験する or 繰り返す. Actually when I saw 「きみとずっと、はじめを」 I instantly recalled this title because its live-action film version was released last month. Well, its actual catchphrase was 何度でも君を思い出す, though.