I'm not sure how well you've parsed/understood the original sentence, so I'll go through it one bit at a time.
In this case, 次 (next) most likely refers to an additional date, given the context of the sentence.
AがBに/と繋がる literally means "A connects to/with B". In your sentence, this is a relative clause describing デート.
Literally "(A) date that connects to the next date". More colloquially, however, we might say "(a) date that leads to another date".
AをBにする translates roughly as "to make A into B". From context, we can reasonably assume that "A" is probably referring to either a current date, or a coming date. This gives this portion the meaning of "(someone) want to make (this date / their upcoming date) into one that leads to another date".
Here, と is the quotative particle, and 誓う translates as "to vow" or "to swear".
So all in all, I would translate your sentence into something like "(Someone) vows to make (their upcoming / this current) date one that will lead to another date".
This says absolutely nothing about whether the speaker intends or doesn't intend to make things serious. We simply cannot know without further context.